Р, мл., П. Грейвс-моррис аппроксимации паде основы теории Обобщения и приложения Перевод с английского Е. А. Рахманова и С. П. Суетина под редакцией А. А. Гончара Москва «Мир» 1986 ббк 22. 13 Б 41



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[Beiker Dzh., Greivs-Morris P. (Baker,Graves-Morri(BookFi)


§ 4.3. Решеточное обрезание в \<р*-евклидовой теории поля 459 Они согласуются также с результатами полученными [Baker et al., 1978] суммированием рядов по Хо методом Паде— Бореля (см. § 1.2). В случае четырех размерностей проведенные вычисления наводят на мысль, что ^0 : ср4:-теория тривиальна (т. е. явление рассеяния отсутствует при любом 0^^о^°°); это оставляет открытым глубокий вопрос о том, каким образом появились превосходные результаты из § 4.2, основанные на теории до перенормировки. Интересная особенность была замечена в случае трех измерений. Предельные переходы при стремлении к нулю периода решетки (ультрафиолето- (ультрафиолетовое обрезание) и при стремлении к бесконечности голой постоянной связи не коммутируют. Следовательно, модель Изинга спин-1/2 нельзя получить с помощью сжатия и предельного перехода в те- теории поля, как это было в случае одной и двух размерностей. Та- Таким образом, Паде-методом в хорошо изученной физической обла- области была обнаружена более богатая структура, чем представлялось ранее.
ЛИТЕРАТУРА Общая литература Abd-Elall L. F.. Delve* L. M.. and Reid J. К. A970). "A numerical method for locating the zeroes and pole* of a meromorphic function." in P. Rabinowiiz. (cd.). Numerical Methods for Sanlincar Algebraic Equations. Gordon and Breach. London, pp. 47-5У. Abramowitz M. and Sicgun I. A964). Handbook of Mathematical Functions. Dover. Aitkcn A. C. A926). "On Bernoulli's numerical solution of algebraic equations." Proc. Ro\\ Soc. Min.. 46.289-305. Aitken. A. C. A964). Determinants and Matrices. Oliver and Boyd. Akhiezer N. I. A965). The Classical Moment Problem. Oliver and Boyd. Alabiso. C. and Butera P. A975). "W-variable rational approximants and method of moments." J. Math. Phys., 16.840-841. Alabiso C. Butera P.. and Prosper:. О. М. A970). "Resolvent operator. Pade approximation and bound states in potential scattering," Lett. Nuovo. Gnu 3,831-839. Alabiso.C, Butera P.. and Prospcri O. M. A971). "Yariational principles and Pad6 approxi- mants. Bound states in potential theory." Nucl. Phys. JS.3I.141-162. Alabiso С Butera. P., and Prosperi G. M. A972a), "Variational principles and Pade approximants. Resonances and phase shifts in potential scattering." Nuct. Phys. B. 42, 493-517. Alabiso C. Butera. P. and prosperi G. M. A972b), "Variational Pade solution of tbc Bethe-Salpeter equation." Nucl. Ph\s. B. 46,593-614. Alder K.. Trautmann. D., and Viollier R. D. A973). "Anwendung der Pade Approximation in der Kernphysifc." Z.fur Naturforschung. 28, 321-331. Alien, G. D.. Chui С. К., Madych W. R.. Narcowich. F. J., and Smith. P. W. A9.74). "Padc approximation and Gaussiar quadrature." Bull. Austr. Math. Soc 10.263-270. Allen G. D., Chui. С. К., Madych W. R.. Narcowich, F. I. and Smith P. W. A975), "Pade approximants of Stieltjes series." J. Approx. Theory, 14,302-316. Allen G. D. and Narcowich. F. J. A975). "On representation and approximation of a class of operator-valued analytic functions," Bull. Am. Math. Soc., 81,410-412. Amos A. T. A978). "Pade approximants and Rayleigh SchrSdinger perturbation theory," J. Phys. B, 11,2053-2060. Anderson G. A740), Letter from Anderson ia Leyden to Jones, in S. J. Rigaud (ed.). Correspondence of Scientific Men of the Seventeenth Camay, George Olms, Hiideshcim A965), Vol. I. p. 361. Argyris J. H,, Yaz. L, E., and William. K. J. A977), "Higher order methods for transient diffuiion analysis," Computer Methods in Appl. Mech. and Engineering 12,243-278.
Литература 461 Arms R. J. and Edrd A. A970), "The Pade tables and continued fractions generated by totally posiiivesequences," in Mathematical Essays, Ohio University Press, Athens, Ohio, pp. 1-21. Axelsson. O. A969), "A class of Л-stable methods," B.I.T., 9, 185-199. Axelsson. O. A972), "A note on a class of strongly //-stable methods," В.1.Т., 12, 1-4. Baker. G. A., Jr. A960), "An implicit, numerical method for solving the н-dimensional heat equation," Quarterly ofAppl. Math., 17, 440-443. Baker. G. A,, Jr. A961) "Application of the Pade approximant method to the investigation of some magnetic properties of the Ising model," Phys. Rev., 124, 768-774. Baker.- G. A, Jr. A965), "The theory and application of the Pade approximant method," Adv. in Theoretical Phys., 1, 1-58. Baker. G. A., Jr. A969), "Best error bounds for Pade approximanis to convergent series of StieItjes."X Math. Phys., 10, 814-820. Baker, G. A., Jr. A970), "The Pade approximant and some related generalizations", in G. A. Baker, Jr., and J. L. Gammel (eds.). The Padi Approximant in Theoretical Physics, Academic Press, New York, pp. 1-39. Baker. G. A., Jr. A972), "Converging bounds for the free energy in certain statistical mechanical problems," J. Math. Phys., 13, 1862-1864. Baker. G. A., Jr. A973a), "Recursive calculation of Pade approximants,"in P. R. Graves-Morris (ed.), Pade Approximates and Their Application;, Academic Press. Baker. G. A, Jr. A973b), "Existence and Convergence of Subsequences of Pade approximants,'* /. Math. Anal. Appl., 43, 498-528. Baker, G. A.. Jr. A974), "Certain invariance properties of the Pade approximant," Rocky Mm. J. Math., 4, 141-149. Baker. G. A, Jr. A975), "Convergence of Pade approximants using the solution of linear functional equations,"./. Math. Phys., 16, 813-822. Baker. G. A., Jr. A976), "A theorem on convergence of Pade approximants," Studies in Applied Math., 55, 107-117. Baker, G. A., Jr. A977), "The application of Pade approximants to critical phenomena," in E. B. Saff and R. H. Varga (eds.), Pade and Rational Approximation, Academic Press, New York, pp. 323-337. .Baker. G. A., Jr. and Chisholm. J. S. R., "The validity of perturbation series with zero radius of convergence,"Л Math. Phys.,7, 1900-1902. Baker,G. A, Jr. and Gammel, J. L. A961), "The Pade approximant," J. Math. Anal. Appl,-2, 21-30. Baker G. A, Jr. and Gammel. J. L A971)', "Application of the principle of the minimum maximum modulus to generalized moment problems and some remarks on quantum field theory," /. Math. Anal. Appl., 33, 197-211. Baker, G. A, Jr., Gammel, J. L., and Wills, J. G. A961), "An investigation of the applicability of the Pade approximant method," J. Math. Anal. Appl., 2, 405-418. Baker. G. A, Jr., Gilbert, H. E, Eve. J., and Rushbrooke, G. S. A967), "High temperature expansions for the spin-^Heisenberg model," Phys. Rev., 164, 800-817. Baker» G. A., Jr. and Graves-Morris P. R. A977), "Convergence of rows of the Pade table," Л Math. Anal. Appl, 57, 323-339. Baker, G. A, Jr. and Gubernatis J. E. A980), "An asymptotic, Pade approximant method for Legendre series," in Proc, 1979 Int. Christo/fel Symposium, cd, P. JL. BuUer, Birkhaubcr Yerlag, (submitted).
462 Литература Baker G. A., Jr. and Hunter D. L. A973), "Methods of series analysis II: Generalized and extended methods with application to the Ising model," Phys. Reo. В 7, 3377-3392. Baker G. A., Jr. and Kincaid. J. M. A979), "The continuous spin Ising model and X:?4:rf field theory," Phys. Rev. Leu., 42, 1431-1434; Errata, 44, 434 A980). Baker. G. A., Jr. and Kincaid. J. M. A980), "The continuous-spin Ising model, g0: 4: d field theory and the renormalization group," J. Stat. Phys., in press. Baker G. A., Jr., Nickel. B. G., Green. M. S., and Meiron. X>. I. A976), "Ising model critical indices in three dimensions from the Callan-Symanzik equation," Phys. Rev. Lett., 36, 1351-1354. Baker. G. A., Jr. and Moussa. P. A978), "Relation in the Ising model of the Lee-Yang branch point and critical behaviour,'"/. Appl. Phys., 49, 1360-1362. Baker. G. A., Jr. and Oliphant. T. A. (I960), "Aniimplicit, numerical method for solving the two-dimensional heat equation," Quarterly of Appl. Math., 17, 361-373. Bareiss E. H. A969), "Numerical solution of linear equations with Toeplitz matrices," Num. Math., 13,404-424. Barnsley. M. V. A973), "The bounding properties of the multipoint Pade approximant to a series of Stieltjcs on the real line,"/. Math. Phys., 14, 299-313. Barnsley. M. F. A976), "Pade approximant bounds for the difference of two series of Stieltjes," J. Math. Phys., 17, 559-565. Barnsley. M. F. and Baker G. A,, Jr. A976), "Bivariational bounds in a complex Hilbert space, and correction terms," J. Math. Phys., 17, 1019-1027. Barnsley. M. F. and Bessis. D. A979), "Constructive methods based on analytic characteriza- characterizations and their applications to non-linear elliptic and parabolic differential equations," J. Math. Phys.. 20, 1135-1145. Barnsley. M. F., Bessi.s D , and Moussa. P. A979), "The Diophantine moment problem and the analytic structure in the activity of the ferromagnetic Ising model,"/. Math. Phys., 20, 535-546. Barnsley. M. F. and Robinson. P. D. A974a), "Pade approximant bounds and approximate solutions for Kirkwood Riscman integral equations,"/. Inst. Math. Appl., 14, 251-285. Barnsley. M. F. and Robinson. P. D. A974b), "Bivariational Bounds," Proc. Roy. Sot: London A 338, 527-533. Barnsley. M. 1". and Robinson P. D. A974c). "Dual variational principles and Pade-type approximants.".Л Inst. Math. Appl.. 14, 229 249. Barnsley. M. F. and Robinson. P. I). A978), "Rational approximant bounds for a class of two variable Stieltjes functions," SIAM J. Math. Anal., 9, 272-290. Basdcvant. J.-L. A969), "Pade approximants," in M. Nikolic (cd.), Methods in Subnuclear Physics, Gordon and Breach, New York, pp. 1-40. Basdevant. J.-L. A972), "Pade approximants in strong interaction physics," in D. Bessis (ed.). Cargese Lectures in Physics, vol. 5, Gordon and Breach. New York, pp. 431-459. Basdcvant. J.-L. A973), "Strong interaction physics and the Pade approximation in quantum field theory," in P. R. Graves-Morris (ed.), Pade Approximants, Inst. of Physics, Bristol, pp. 77-100. Basdevant, J.-L., Bessis. D., and Zinn-Justin. J. A968), "Pade approximants in strong interac- interactions. Two body pion and Kaon systems," Phys. Lett., 27B, 230-233. Basdcvant, J.-L., Bessis D., and Zinn-Justin, J. A969), "Pade approximants in strong interac- interactions. Two body pion and Kaon systems," Nuovo Ст., 60А, 185-238.
Литература 463 Bcssis. D. and Pusterla. M. A968), "Unitaiy Pade approximants in strong coupling field theory and application to the p and/0 meson trajectories," Nuovo Cim., 54A, 243-294. Bcssis D. and Turchctti.,G. A972), "Renormalization of the a-model through Ward identities," in Carghe Lectures in Physics, Vol. 5, ed. D. Bessis, Gordon and Breach, New York, 119-178. Bcssis, D. and Turchetti. G. A977), "Variational matrix Pade approximants in potential scattering and low energy Lagrangian field theory," Nucl. Phys. Bx 123,173-188.- Bcssis. D. and Villani M. A975), "Pcrturbativc-variational approximations to the spectral properties of semi-bounded Hilbert space operators, based on the moment problem with finite or diverging moments. Application to quantum mechanical systems," J. Math. Phys., 16, 462-474. Biswas. S. N. and Vidhani, T. A973), "Solution of Schr5dinger equation with continued fractions,"/. Phys. A, 6, 468-477. Blanch. O. A964), "The numerical evaluation of continued fractions," SI AM Rev., 6,383-421. Bombelli, R. A572), L'Algebra, Yenezia. Boas. R. P. A954), Entire Functions, Academic Press, New York. Brent, R. P., Gustavson. F. G. and Yun,.D. Y. Y. A980), "Fast solution of Toeplitz systems of equations and computation of Fade approximants," /. Algorithms, 1, 259-295. Brczia E, Le Guillou J. C, and Zinn-Justin. J. A977), "Perturbation theory at large order. I The фг" interaction," Phys. Rec. D, 15, 1544-1558. Brezinski C, A97)), "Acceleration dc suites a convergence logarithmique," Comptes Rendus, 273A, 727-730. Brezinski, С A972), "Conditions d'application et de convergence de precedes d'extrapolation," Num. Math., 20, 64-79. BrczinskLC. A974), "Review of acceleration methods," Rendicont! di Mat.,1,303-316. Brezinski. C. A976), "Computation of Pade approximants and Continued fractions,"/. Сотр. Appl. Math.,2, 113-123. Brezinski.. C. A977), Acceleration de la Convergence en Analyse Numerique, Springer Lecture Notes in Mathematics, No. 584, Berlin. Breziaski, C. A978a), "Convergence acceleration of some sequences by the e-algorithm," Num. Muth.,29, 173-177. Brezinski. C. A978b), "Pade type approximants for double power series,"/, Indian Math. Soc, 42, 267-282. Brezinski. С A979), "Rational approximation to formal power series," J. Approx. Theory, 25, 295-317. Brezinski C. A980), "A general extrapolation algorithm," Num. Math., 35, 175-187. Bre/inski. C. and Rieu/ A. C. A974), "Solution of systems of equations using the ^-algorithm, and application to boundary value problems," Math. Сотр., 28,731-734. Bulirsch. R. and Stoer. J. A964), "Fchler abschatzungen und Extrapolation mit rationalen Functionen bei Yerfahren vom Richardson-Typus," Num. Math., 6, 413-427. Bultbeel, A. A979), "Rccurisivc algorithms for the Pade table: two approaches," in L. Wuytack (ed.); Pade Approximation ami Its Applications, Springer Lecture Notes in Mathematics, No. 765, pp. 211-230. Bultbeel. A. A980a), "Recursive algorithms for the matrix Pade problem," Math. Сотр., 35, 875.-892.
464 Литература Basdevant. J.-L. and l.ee. B. W. A969a), "Pade approximants in Ihe Литература 465 Buliheel, A. A980b), "Dhision algorithms for continued fractions and the Pade table," J. Сотр. Apphl. Math,, 6, 259-266. Bulthcel A. and Wuytack,. L., A981), "Stability of numerical methods for computing Pade approximants", in Approximation Theory 111, cd, E. W. Cheney, Academic Press, New York. Bus, J. C. 1'. and Dckker T. J. A975), Two efficient algorithms with guaranteed convergence for finding, a zero of a function," A. C. M. Trans, on Math. Software, 1, 330-345. Butcher. J. С A963), "Coefficients for the study of Rungc-Kutla integration processes," J. Australian Math. Soc.,X 185-201. Butcher J. С A964), "Implicit Runge-Kutta Processes." Math. Сотр., 18, 50-64. Carlcman. T. A926), Les Functions Quasi-Anafytiques, Gauthier Villars, Paris; translated by J. L. Gammel, Los Alamos Tech. report 4702, A971). Carroll, A., Baker G. A,, Jr,, and Gammel. J. L. A977), "A comment on a paper of Hamer applying Pade approximants to a 1 +1 dimensional lattice model of Q.C.D., Nucl. Phys. B. 129, 361-364. Carslav. H. S. and Jaeger J. C. A959), Conduction of Heat in Solids, Oxford University Press. Cartan. H. A928). "Sur les systemes de fonctions holomorphes a vari&tes lineaires ct leurs applications." Ann. Sci. Ecole. Norm. Sup. C), 45,255-346. Caser. S.. Piquet C, and Vermeulen. J. L. A969), "Pade approximants for a Yukawa potential." Nucl. Phys. B, 14, 119-132. Cavendish. J. С Culham. W. E. and Varga, R. S. A972), "A comparison of Crank-Nicholson and Chcbychev rational methods for numerically solving linear parabolic equations," Л Сотр. Phys., 10,354-368. Cauchy. M. A.-L. A821). "Cours d'analyse" de l'Ecole Royale Polytechnique; premiere partie, L'lmprimerie Royale, Paris. Char B. W. A980), "On Sticlrjes continued fraction for tbe Gamma Function," Math. Сотр., 34, 547-551. Cheney E. \V. A966), Introduction to Approximation Theory, McGraw-Hill. Chipman. F. H. A971), "Л-stable Runge-Kutta processes," B.I.T., 11,384-388. Chisholm, J. S. R. A963), "Solution of linear integral equations using Pad& approximants,"/. Math. Phys., 4. 1506-1510. Chisholm, J. S. R. A966), "Approximation by sequences of Pade approximants in regions of mcromorphy," J. Math. Phys., 1, 39-46. Cliisholm, J. S. R. A973), "Rational approximants defined from double power series," Math. Сотр., 27. 841-848. Chisholm J. S. R. A977a). "ЛГ-variable rational approximants," in Pade and Rational Ap' proximation, eds. E. B. Saff and R. H. Varga, Academic Press, New York, 23-42. Oiisholm J. S. R. A977b), "Multivariate approximants with branch points. I. Diagonal approximants," Proc. Roy. Soc. Loud. A, 358. 351-366. Chisholm. J. S. R. A978a). "Multivariate approximants with branch cuts," in Multivariate Approximation, cd. D. C. Handscomb, Academic Press, London, Chisholm. J. S. R. A978b), "Multivariate approximants with branch points. II. Off-diagonal approximants," Proc. Roy. Soc. Land. A, 362, 43--56. Chisholm, J. S. R. and Common, A. K. A980), "Generalizations of Pade approximation for Chcbychev and Fourier series," in Proc. 1979 Int. Christoffel Symposium, ed. P. L. Butzer, Bifkhauser Verlag.
466 Литература Chisholm. J. S. R., Genz A. C, and Rowlands, G, E. A972), "Accelerated convergence of sequences of quadrature approximations,"/. Сотр. Phys., 10, 284-307. Chisholm. J. S. R. and Graves-Morris. P. R. A975), "Generalizations of the theorem of dc Montessus to two-variable approximants," Proc. Roy. Soc. Lond. A, 342, 341-372. Chisholm, J. S. R. and Hughes Jones. R. A975), "Relative scale covariance of ^-variable approximants," Ргж. Roy. Soc. Lond. A, 344, 465-470. Chisholm, J. S. R. and McEwan J. A974), "Rational approximants defined from power Series in .V-variables," Proc. Roy. Soc. lond. A. .136, 421-452. Chisholm. T. S. R, and Roberts. D. E. A976), "Rotationally covariant approximants derived from double power series," Proc. Roy. Soc. Lond. A, 3S1, 585-591. Christoffcl, E. B. A877). "Sur une classe particuliere de fonctions entieres et de fractions continues," Ann, ili Mat., 8, 1-10. Chui С. К., Shisha. О., and Smith. P. W. A974), "Pade approximanls as limits of best rational approximants," Л Approx. Theory, 12, 201-204. Chui. С. К., Shisha. О., and Smith. P. W. A975), "Best local approximation," /. ApproX. Theory, 15, 371-381. Chui ,C. K., Smith, P. W., and Ward. J, D. A978), "Best L2 local approximation," /. Approx. Theory, 22, 254-261. Clacssens. G. A976), "A new algorithm for osculatory rational interpolation," Num. Math.,27, 77-83. Clacssens, G. A978a), "On the Newton-Pade approximation problem," J. Approx. Theory, 22, 150-160. Clacssens. G. A978b), "On the structure of the Newlon-Pade table," /. Approx. Theory, 22, 304-319. Claesscns. G. A978c), "Generalized e-algorithm for rational interpolation," Num. Math., 29, 227-231. Claessens. G. and Wuytack. L. A979), "On the computation of non-normal Fade approxi- approximants,"./. Сотр. Appl. Math., 5, 283-289. Cknshaw С W. and Lord. K. A974), "Rational approximations from Chcbychcv series," in Studies in Numerical Analysis, ed. В. К. P. Scaife, Academic Press, London, 95-113. Cody. W. J., Meinardus. G., and Varga R. S. A969), "Chebychev rational approximations to e~x in [0,oo) and applications in heat conduction equations," J. Approx. Theory, 2, 50-65. Common. A. K. A968), "Padc approximants and bounds to series of Stieltjcs,"/. Math. Phys., 9, 32-38. Common. A. K. A969a), "Properties of tegendre expansions related to Stieltjcs series and applications to ir-ir scattering," Nuovo Cim., 63A, 863-891. Common, A. K. A969b), "Properties of generalisations to Pade approximants," J. Math. Phys., 10, 1875-1880. Common. A. K. A970), "Some consequences of relations of it—n partial wave amplitudes to Stieltjcs series," Nuovo Cim., 6SA, 581-596. Common. A. K. A979), "Calculation of Yukawa scattering amplitude and impact parameter amplitudes using Legendre-Pade approximants,"/. Phys. A, 12, 2563-2572. Common A. K. and Graves-Morris, P. R. A974), "Some properties of Chisholm approximants,'* J. Inst. Math. Appl., 13, 229-232.
Литература 467' Common А. К. and Stacey Т. W. A979a), "The convergence oF Legendre-Pade approJtimant» to.the Coulomb and other scattering amplitudes," /. Phys. A, 11,275-289. Common, A. K. and Stacey, T. W. A979b), "Legendre-Pade approximates and their applb cation in potential scattering,"/. Phys. A, 11,259-273. Common A, K. and Stacey. T. W. A979c), "Convergent series of Legendre-Fade approxi» mants to the real and imaginary parts of the scattering amplitudes," J. Phys, A, 12, 1399-1417/ Conn. U. W. A974), "Higher order variational principles and Pade approximants for linear functional," Nucl. Scl. Ens;., 55, 468-470. Copley. L. A., Elias, D. K., and Masson. D. A968), "Broken symmetry model of mesoil interactions," Phys. Re»., 173, 1552-1563. Copley L. A. and Masson, I). A967), "Pade approximant calculation of ff-JT scattering," Phvs. Rev., 164, 2059-2062. Copson E. T. A948), An Introduction to the Theory of a Complex Variable, Oxford University Press. Corcoran С. Т. and Langhoff. P. W. A977), "Moment theory approximations for non-negative spectral densities," /. Math. Phys., 18, 651-657. Cordellier. F. A979a), "Sur la rcgularite des precedes S* d'Aitken ct W de Lubkin," in L. "Wuytack (ed.), Pade Approximation and Its Applications, Springer Lecture Notes in Math., No. 765, Berlin. Cordeliier, F. A979b), Demonstration algcbrique. de 1' extension de l'idcntite de Wynn au? tables de Pade non-normales," in L. Wuytack (ed), Pade Approximation and its Applications, Springer Lecture Notes in Math., No. 765, Berlin. Courant R. and Ifflbert D. A953), Methods of Mathematical Physics, Vol. I, Intcrscicnce. Crank. J. A975), The Mathematics of Diffusion, Oxford University Press. Dekker. T. J: A969), "Finding a zero by means of successive linear interpolation," in Constructive Aspects of the Fundamental Theorem of Algebra, eds. B. Dcjon and P. Hcnrici, J. Wiley, 37-48. Delves. L. M. and Phillips, A, C. A969), "Present status of the nuclear three body problem." Reo. Mod. Phys., 41, 497-530. Dennis, J. J. and Wall H. S. A945), "The limit-circle case for a positive definite ./-Ггаси'опЛ Duke Math. J., 12,255-273. Dienes. P. A957), "Hie Taylor series," Dover, New York. Dijkstra D. A977), "A continued fraction expansion for a generalization of Dawson's integral," Math. Сотр., 31, 503-5Ю. Dimock. J. A974), "Asymptotic perturbaiion expansion in the Р(Ф)^ quantum field theory," Coiunt. Math. Phys., 35, 347-356. Dirae P. A. M. A958), "Principles of Quantum Mechanics," Oxford University Pres!>. Domb С A974), "Ising model," in Phase Transitions and Critical Phenomena" vol. 3, eds, С Domb and M. S. Green, Academic Press, London, 357-485. Domb C. and Sykes. M. F, A961), "Use of series expansions for the Ising model susceptibility and excluded value problem,"/. Math, Phys., 2, 63-67. Donnelly J. D. P. A966), "The Pade table," in Methods of Numerical Approximation, ed. D. C. Handscomb, 1'ergamon Press. Oxford, 125-130. Drew, D. and Murphy J. A. A977), "Branch points, ,U-fractions and rational approxhnantS- generated by linear equations,"/, lust. Math. Appl., 19, 169-185.
¦468 Литература Dyson F. J. A952). "Divergence of Perturbation Theory in Quantum Electrodynamics," Phys. Rev. 85. 631-632. Eekmann. J.-P., Magnen J., and Seneor. R. A974), "Decay properties and Borel summability for the Schwinger functions in Р(ф)г theories," Comm. Math. Phys., 39, 251-271. Edrei A. A939), "Surles determinants rtcurrentsetlessingularitesd'unefonctiondonnees par son developpement de Taylor," Compositio Math., 7,20-88. Edrei. A. A953), "Proof of a conjecture of Schoenberg on the generating function of a totally positive sequence." Can. J. Math., 5, 86-94. Edrei. A. A975a), "Convergence of complete Pade tables of trigonometric functions," J. Approx. Theory, 15, 278-293. Edrei. A. A975b), "The Pade table of functions having a finite number of essential singulari- singularities," Pacific J. Math., 56,429-453. Ehle. B. L. A968). "High order Л-stable methods for numerical solution of systems of differential equations," B.I.Т., 8, 276-278. Ehle. B. L. A973), "/(-stable methods and Pade approximants to the exponential," SIAMJ. Math. Anal., 4, 61 \-680. .Ehle. B. L. A976), "On certain order constrained Chebychev rational approximations,"/. Approx. Theory, 17. 297-306. Erdelyi. A. A956), Asymptotic Expansions, Dover, New York. Euler. L. A737), "De fractionibus continuis," Comm. Acad. Set. Tmper. Petropol., 9. Fair. W. G. A964), "Pad6 ajiproximants to the solution of the Rlcatti equation," Math. Сотр., 18.627-634. Fairweather G. A971), "A survey of discrete Galerkin methods for parabolic equations in one space variable," Math. Colloq. U.C.T., 7,43-77. Fcldman. J. S. and Osterwalder. K. A976), "The V/ightman axioms and the mass gap for weakly coupled (4K quantum field theories," Amu Phys., 97, 80-135. Ferer. M., Moore. M. A., and Wortis. M. A971), "Some critical properties of the nearest- neighbor, classical Heisenberg model for the f.c.c, lattice in finite field for temperatures greater than 7^.," Phys. Rev. B, 4, 3954-3963. Ferrar. W, L. A938), Convergence, Oxford University Press. Fisher; M. E A977), "Scries expansion approximants for singular functions of many variables," in Statistical Mechanics and Statistical Methods In Theory and Application, ed. Uzi Landman, Plenum Press, 3-31. Fisher. M. E. and Kerr R. M. A977), "Partial differential approximants for multicritical singularities," Phys. Rev. Lett., 39,667-670. Fleischer. J. A972), "Analytic continuation of scattering amplitudes and Pade approximants," Nucl. Phys. B, 37, 59-76. Fleischer J. A973a), "Nonlinear Pade approximants for Legendre series," J. Math. Phys., 14, 246-248. Fleischer J. A973b), "Generalizations of Pad6 approximants," in Pade Approximants, ed. P. R. Graves-Morris, Inst. of Phys., Bristol, 126-131. Fleischer J., Gammel J, L., and Menzel, M. T. A973), "Matrix Pade approximants for the 'Sa and 3P0 partial waves in nucleon-nucleon scattering," Phys. Rev. D, 8, 1545-1552. Fleischer, J. and Tjon. J. A. A975), "Bethe-Salpeter equation for /=0 nucleon-nucleon scattering with one boson exchange," Nucl. Phys. B, 84,375-396. -Fleischer. I. and Tjon, J. A. A980), "Bethe-Salpeter equation for clastic nucleon-nucleon scattering," Phys. Reo. D, 21, 87-94.
Литература 469 I'ogli. G. L., Pcllicoro M. F., and Villani. M. A971), "A summation method lor a Class o[ series with divergent terms," Nuooo Cim., 6A, 79-97. Fogli. G. L, Pellicoro M. F., and Villani M. A972), "An approach to the radiative corrections in Q.E.D. in the framework of the Pade method," Nuavo Cim., HA, 153-177. Ford. W. B. A960), Asymptotic Series and Divergent Series, Chelsea, New York. Frank. W. M., Land. D. I, and Spector. R. M. A971), "Singular potentials," Rev. Mod. №x, 43,36-98. Fralamico. G., Ortolan! F, and Turchetti, G. A976) "Exact solutions from the variational [1/1] matrix Pade approximant in potential scattering," Lett. Nttovo Cim,, 17, 582-584. Frnbenius, G. A881), "Ucbcr Rclationen zwischen den Naherungsbrlkhcn von Potcnzreilien," J.fur Rcine unit Angewandte Math., 90, 1-17. Gallucci, M. A. and Jones W. B. A976), "Rational approximations corresponding to Newton Series (Newton-Pade approximants),"/. Approx. Theory, 17, 366-392. Oammel, J. L. A973), "Review of two recent generalizations Of the Pade approximant," in Pade Approximates and their Applications, ed. P. R. Graves-Morris, Academic Press, London, 3-9. Caramel, J. L. and McDonald. F. A. A966), "Applications of the Pade approximant to scattering theory," Phys. Reo., 142, 1245-1254. Gammel. J. L., Rousseau, С. С, and Saylor. D. P. A967), "A Generalization ot the Pade, Approximant," Л Math, Anal, Appl, 20, 416-420. Gargantini. J. and Henrici, P. A967), "A continued fraction algorithm for the computation' of higher transcendental functions in the complex plane," Math, Сотр., 21, 18-29. Oaribotti. C. R. A972), "Sehwinger variational principle and Padc approximants,"^wi. Phys., 71,486-496. X O.nibotti. С R. and Grinstein. F. F. A978a), "Summation of partial waves fof long range potentials V'J. Math. Phys., 19, 821-829. Gaubotti. С R. and Grinstein. F. F. A978b), "Punctual Pade approximants as a regularization procedure for divergent and oscillatory partial wave expansion of the scattering ampli- amplitude," J. Math. Phys., 19, 2405-2409. G.iribottL C. R. and Grinstein. F. F. A979), "Summation of partial waves for long range potentials II," Л Math. Phys., 20, 141-147. Garibotti. С R., Pcllicoro. M. F., and Villani. M. A970), "Pade method in singular potentials," Nuovo Cim., 66A, 749-766. Ciaribotti, С R. and Villani. M. A969a), "Continuation in the coupling constant for the total Г and A' matrices," Л'шх-о Cim., S9A, 107-123. Garibotti, C. R. and Villani, M. A969b), "Padc approximant and the Jost function," Ntioio Cim., 6IA, 747-754. Garnctt. J. A973), Analytic Capacity and Measure, Springer Lecture Notes in Mathematics, No. 297. Oarside, G. R., Jarratt. P., and Mack. C. A968), "A new method for solving polynomial equations," Сотр. J., 11, 87-90. ¦Gauss, С F. A813), "Disquisitiones gcnerales circa seriam infiniiam I-Y l-2y(y+l) Coumnwaiiont's i'wiV(«».i Ri^iai' Swuliviwt OwVii'ijjauM RiWtUWKS, 2,
470 Литература Ciauthier. P. M. A977), "On the possibility of rational approximation." in Padk and'rational approximation, eds. E. B. Saff and R. S. Varga, Academic Press, .Vew Vorfc. pp. 261-264. CJautschi. W. A967), "Computational aspects of three term recurrence relations," SIA\t Review, 9, 24-82. Gear С W. A971), Numerical Initial Value Problems in Ordinary Differential Equations. Prentice Hall. Geddes. K. 0. A979), "Symbolic computation of Fade approximants." A.C.M. Trans. Math. Software, 5, 218-233. Gekeler. E. A972), "On the solution of systems of equations by the epsilon algorithm of Wynn," Math. Сотр., 26, 427-436. Genz. A. C. A972), "An adaptive multidimensional quadrature procedure." Сотр. Phys. Comm., 4, 11-15. Genz A. C. A973), "The applications of the f-algorithm to quadrature problem»." in Padb Approximates, ed. P. R. Graves-Morris. Inst. of Physics. Bristol. 105-116. Genz A. C. A974), "Some extrapolation method* for th'e numerical calculation of multidimen- multidimensional integrals," in Software for Numerical Mathematics, cd. D. J, Evan». Academic Press, New York, 159-172. Genz. A. C. A977). "A non-iiuear method for the convergence of multidimensional sequences.'* /. Comp.Appt. Math.,3, 181-184. Germain-Bonne, B. A979), "Ensembles des suites et de precedes lies pour I'ucwlcration dc la convergence." in L. Wuytack (ed.) Pedi Approximation and Its Applications, Springer Lecture Notes ia Mathematics, No. 765, pp. 116-134. Gexsten. A,, Owen, D. A., Gammel. J. L., Mery. P,, and Turchetti. O. U976). "Matrix PadS approximants and the Bcthe-Salpeler equation of the nueleon-nucleon interaction," Phys. Лее. D. 13. 1140-1143. Cilewicz J. A978). Approximants tie Pude, Springer Lecture Notes in Mathematics, No. 667. Giraud, В. С A978). "Lower bounds to bound state eigenvalues," Phys. Rev. С 17.800-809. Giraud B. G., Klialil. А. В., and Moussa, P, A976), "Pade approximants for distorted waves." Phys.Reo, C. 14, 1679-1687. Giraud. B. G. and Turchetti G. A978), "Rigorous lower bounds to the energy levels with finite summations," Lett. Nuovo Cim., 21, 605-608. Coldberger M. L. and Watson. К. М. A964), Collision Tlieory. J. Wiley. New York. Cioldhammer P. and Feenberg E. A956), "Refinement of the Brillouin Wigncr perturbation method," Phys. Reo., 101. 1233-1234. Goncar. A. A. and Lopez Guilkrmo t. A978), Math. U.S.S.R. Shornik, 34,449-459. Cordon. R. G. A968), "Error bounds in equilibrium statistical mechanics," Л Math. Phys ¦ f. 655-663. Ciitz. U., Robcl. F., Trautmann, D., and Jocliin H. A976), "Inuucnce of the Mott-Schwingcr interaction on the elastic scattering of protons," Z. Phys. A, 278. 139-143. Graffi. S., Grccchi V.. and Simon. B. A970), "Borel summability: application to the anharnioiiic oscillator." Phys. Lett., 32B, 631-634. Gtaffi. S. and Grccchi V. A978), "Borcl summability and indeterminacy of the Stielt'> moment problem: application to the (inharmonic oscillators." J. Math. Phys.. i^, 1002-1006. Graff' S-. Grecchi. V., and Turchctli. G. A971). "Summation methods for the perturbation series of the generalized, harmonic ostillaior," Know Cim., 4B, 313-340.
Литература 471 Gragg. W. В. A965), "On extrapolation algorithms for ordinary initial value problems," SIAM J. Num. Anal., 2, 384-403. Gragg. W. B. A968), "Truncation error bounds for ^-fractions," Num. Math., 11, 370-379. Gragg W. B. A970), "Truncation error bounds for jr-fractions," Bull. Am. Math. Soc., 76, 1091-1094. Gragg W. B. A972), "The Pade table and its relation to certain algorithms of numerical analysis," SIAM Review, 14, 1-62. Gragg. W. B. A974), "Matrix interpretations and applications of the continued fraction algorithm," Rocky Mtn. J. Math., 4, 213-225. Gragg W. B. A977), "Laurent, Fourier and Chebychev Pade tables," in Padeand Rational Approximation, cds. E B. Saff and R. H. Varga, 61-70. Gragg, W.'B. and Johnson. G. D.' A974), The laurent fade Table, Info. Proc. 74, North Holland, Amsterdam, 3, 632-637. Graves-Morris, P. R. A973), "Pade approximants and potential scattering," in Pade Approxi- mants, ed. P. R. Graves-Morris, Inst. of Phys., Bristol, 64-76. Graves-Morris P. R. A975), "Convergence of rows of the Padfc table," in H. Cabannes (ed.), Pade Approximation and Its Application to Mechanics, Springer Lecture Notes in Physics, No. 47, pp. 55-68. Graves-Morris. P. R. A977), "Generalisations of the theorem of de Montessus using Canter- Canterbury approximanls," in E. B. Saff and.R. S. Varga (eds.), Pade and Rational Approxima- Approximation, Academic Press, New York, pp. 73-82. Graves-Morris. P. R. A978a), "Pade approximants for integral equations?,"/. Inst. Math. Appl., 21, pp. 375-378. Graves-Morris. P. R. A978b), "Applications of matrix Padc approximants in potential theory," Ann. Phys., 114, 290-295. Graves-Morris. P. R. A979), "The numerical calculation of Pade approximants," in L. Wuytack (ed.), Pade Approximation and Its Applications, Springer Lecture Notes in Mathematics, No. 765, pp. 231-245. Graves-Morris, P. R. A980a), "Practical, reliable, rational interpolation," J. Inst. Math. Appl., 25,267-286." . Graves-Morris. P. R. A980b), "A generalized Q.D. algorithm," /. Сотр. Appl. Math., 6, 247-249. Graves-Morris: P. R. A981) "The Convergence of Ray Sequences of Pade Approximants of Stieltjes Series" (preprint). Graves-Morris. P. R. and Hopkins. T. R. A981), "Reliable rational interpolation," Num. Math.,36, 1.11-128. Graves-Morris, P. R., Hughes Jones. R., and Makinson,G. J. A974), "The calculation of some rational approximants in two variables,"/. Inst. Math. Appl., 13, 311-320. Graves-Morris. P. R. and Rennison. J. F. A974), "Analyticity in the coupling strength," J. Math. Phys., 15, 230-233. Graves-Morris. P. R. and Roberts, D. E. A975), "Calculation of Canterbury approximants," Сотр. Phys. Comm., 10, 234-244; Erratum, A977), 13, 72. Graves-Morris P. R. and Samwell. С J. A975), "Canterbury approximants in potential scattering,"/. Phys. C, 1, 805-814. Gray. H. L., Atchison. T. A., and Me Williams,G. V. A971), "Higher order C-transformations," SIAMJ. Num. Anal., 8, 365-381.
472 Литература Grenandcr, U. and Szeg6, G. A958), Toeplilz Forms and Their Applications, Univ. of California Press, Berkeley. Grinstein. F. F. A980), "Summation of partial wave expansions in the scattering by short range potentials," J. Math. Phys., 21, 112-119. Guttmann. A. J. A969), "Determination of critical behaviour in lattice statistics from series expansions HI," J. Phys, C, 2, 1900-1907. Guttrnan, A. J. A975a), "On the recurrence relation method of series analysis,"/. Phys. A, 8, 1081-1088. Guttmann, A. J. A975b), "Derivation of 'mimic expansions' from regular perturbation expan- expansions in fluid mechanics," J. Inst. Math. Appl., 15, 307-315. Guttmann. A. J. and Joyce G. S. A972), "On a new method of series analysis in lattice statistics," J. Phys. A, 5, L81-L84. • Hadamard, J. A892), "Essai sur l'etude des fonctions donnees par leur developpement de Taylor," 2lbm° partie, J. de Math., 4, 101-186. Hadamard J. (968"), Oeuvres, Vol. 2, CNRS, Paris, pp. 24-60. Hamburger. H. A920), "Ueber eine Erweiterung des Stieltjes'schen Momentenproblems," Math.Armaleii, 81, 235-319. Hamburger. H. A921), "Ueber eine Erweiterung des Stieltjes'schen Momentenproblems," Math. Annalen, 82, 120-164, 168-187. Hardy G. H. A956), "Divergent Series," Oxford University Press. Haymaker. R. W. A968), "Application of analyticity properties to the numerical solution of the Bethe-Salpeier equation," Phys. Rev.. 165, 1790-1X02. Haymaker. R. W. and Schlessinger L. A970), "l'ade approximants as a computational tool for solving the Schrodinger and Bethe-Salpeter equations," in The Pade Approximant in Theoretical Physics, eds. G. A. Baker, Jr. and J. I.. Gammd, Academic Press, New York, 257-288. Henrici. P. A958), "The quotient difference algorithm," \at. Bur. Stand.—Appl. Math Series, 49,23-46. Henrici. P. A964), Elements of Numerical Analysis, Wiley. Henrici. P. A974). Applied and Computational Complex Analysis, Vol. J, Wiley. Henrici. P, and Pfluger P. A966), "Truncation error estimates for Stieltjes fractions," Num. Math. ,9, 120-138. Hildebrand, F. B. A956), "Introduction to Numerical Analysis," McGraw-Hill. Hillc K. A959), Analytic Function Theory, Vol. I, Ginn and Co. Hille F.. A962), Analytic Function Theory, Vol. II, Ginn and Co. Hillion. P. A977a), "Remarks on rational approximation of multiple power series," /. Inst. ¦ Math. Appl., 19.281-293. Hillion. 1'. A977b), "Approximating functions with a given singularity," J. Math. Phys., 18, 465-470. Hofman II. M., Starkand. Y., and Kirson. M. W. A976), Nucl. Phys., A 266, 138^162. Holdeman, J. Т., Jr. A969), "A method for the approximation of functions defined by formal series expansion in orthogonal polynomials," Math. Сотр., 23, 275 -287. Householder, A. S. A970), The Numerical Treatment of a Single Non-linear liquation, McGraw- Hill. New York. Householder A. S. A971), "The l'ade table, the Frobcniub identities and the q.d. algorithm," Lin. Algebra and Its Appl.. 4, 161-174.
Литература 473 Householder A. S. and Stewart G. W., Ш A969), "Bigradients, Hankel determinants and the Pade table," in B. Dejon and P. Henrici (eds.), Constructive Aspects of the Fundamental Theorem of Algebra, Academic Press, New York, pp. 131-150. Hughes. Jones R. A976), "General rational approximants in /V-variables,"/. Approx. Theory, 16,201-233.- Hughes. Jones R. and Makinson. G. J. A974), "The generation, of Chisholm rational approxi- approximants to power series in two-variables," /. Inst. Math. Appl,, 13,299-310. Hunter. D. L. and Baker. G. A., Jr. A973), "Methods.of series analysis I: comparison of methods used in critical phenomena," Phys. Rev. B, 7, 3346-3376.. Hunter. D. L. and Baker, G. A., Jr. A979), "Methods of series analysis HI: integral approximant methods," Phys. Rev. B, 19, 3808-3821. Ince E. L. A944), Ordinary Differential Equations, Dover Publications, New York. Isaacson. E. and Keller H. B. A966), Analysis of Numerical Methods, J. Wiley. Isenberg. С A963), "Moment calculations in lattice dynamics I, F.C.C. lattice with nearest -neighbor interactions," Phys. Rev., 132, 2427--2433. Iserles A. A979), "A note on Pade approximations and generalized hypergeometric functions," B.I.T., 19, 543-545. Jacobi. C. G. J. A846), "Ober die Darstellung einer Reihe gegebner Werthe durch eine gebrochne rationale Function," J. fur Relne Atigewandte Math., 30, 127-156. Jarratt P. A970), "A review of methods for solving non-linear algebraic equations in one variable," in Numerical Methods for Non-linear Algebraic Equations, ed. P. Rabinowitz, Gordon and Breach, London, 1-26. Jones W. B. A977), "Multiple point Pade tables," in E B. Saff and R. H. Varga (eds.), Pade and Rational Approximation, Academic Press, New York, pp. 163-172. Jones W. B. and Thron^W. J. A966), "Further properties of Г-fractions," Math. Aimalen, 166, 106-118. Jones W. B. and Thron. W. J, A971), "A posteriori bounds for the truncation error of continued fractions," SIAMJ. Num. Anal., 8, 693-705. Jones. W. B. and Thron W. J. A974), "Numerical stability in evaluating continued fractions," Math. Сотр., 28, 795-810. Jones. W. B. and Thron W. J. A975), "On the convergence of Pade approximants," SIAMJ. Math.Anal.,6,9-\6. Jones W. B. and Thron. W. J. A977), "Two point Pade tables and Г-fractions," Bull. Am. Math. Soc, 83, 388-390. Jones, W. B. and Thron W. J. A979), "Sequences of me'romorphic functions corresponding to formal Laurent series," SIAMJ. Math. Ana!., 10, 1-17. Jones, W. B. and Thron W. J. A980), Continued Fractions, Analytic Theory and Applications, Addison-Wesley, Reading, Mass. Jones. W. В., Thron W. J., and Waadeland. H. A980), "A strong Stieltjes moment problem," Trans. Am. Math. Soc, 261, 503-528. Joyce D. С A971), "Survey of extrapolation processes in numerical analysis," SIAMReo., 13, 435-490. Joyce G. S. and Guttmann A. J. A973), "A new method of series analysis," in Padi Approximants and their Applications, ed. P. R. Graves-Morris, Academic Press, London, 163-167. iCananer. D. K. A972), "Numerical quadrature by the e-algorithm," Май. Сотр., 26, 689-694.
474 Литература Kailath. Т., Vieira. A., and Morf, M. A978), "Inverses of Toeplitz operators, innovations and orthogonal polynomials," SIAM Review, 20, 106-119. Karlin S. A968), Total Positivity I, Stanford Univ. Press. Karlsson, J. A976), "Rational interpolation and best rational interpolation," /. Math. Anal. Appl., 53, 38-52. Karlsson, I and von Sydow, B. A976), "The convergence of Pade approximants to series of Stieltjes," Arkivfor Matematik, 14, 43-53. Karlsson. J. and Wallin, H. A977), "Rational approximation by an interpolation procedure in several variables," in Pade and Rational Approximation,' eds. E. B. Saff and R. H. Varga, Academic Press, New York, 83-100. Khalil, A. B. A977), "The ЛГ-matrix in distorted wave theory," Nuovo Cim., 36A, 354-366. Killingbeck, J. A977), "Quantum mechanical perturbation theory," Rep. Pros. Phys., 40, 963-1031. Killingbeck, J. A980), "The harmonic oscillator with \xm perturbation," J. Phys. A, 13,49-56. Kogbetlianz, E. G. A960), "Generation of elementary functions," in A. Ralston and H. S. Wilf (eds.), Mathematical Methods for Digital Computers, Wiley, pp. 7-35. Kronecker, L. A881), "Zur Theorle der Elimination einer Variabeln aus zwci algebi aischen Gleichungen," Monatsberlchte der Koniglich Preussichen Akademie der WissensAaftemu Berlin, 535-600. Laguerre. E. A879), "Sur la reduction en fractions continues d'une fonction qui satisfait a une equation lineaire du premier ordre a coefficients rationnels," Bull. Soc. Math de France, 8, 21-27. Laguerre. E A885), "Sur la reduction en fractions continues d'une fraction qui satisfait a une equation differentielle lineaire du premier ordre dont les coefficients sont rationnels," /. de Math, Pures et Appliques, I, 135-165. Lambert, J. D. A974), "Two unconventional classes of methods for stiff systems," in R. A. Willoughby (ed.), Stiff Differential Equations, pp. 171-186. Lambert J. D. A975), "Computational methods in ordinary differential equations," J. Wiley. Lambert J. D. and Shaw, В., A965), "On the numerical solution of y'~J(x,y) by a class of formulae based on rational approximation," Math. Сотр., 19, 456-462. Lambert J. D. and Shaw, B. A966), "A generalization of multistep methods for ordinary differential equations," Num. Math., 8, 250-263. Lanczos, C. A952), "Solution of systems of linear equations by minimized iterations," /. Res. of Nat. Bur. Stan., 49, 33-53. Lanczos,. C. A957), "Applied Analysis," Pitman Press. Larkin. F. M. A967), "Some techniques for rational interpolation," Сотр. J., 10, 178-187. Larkin. F. M. A980), "Root finding by fitting rational fractions," Math. Сотр., 35, 803-816. Lee, T. D. and Yang, С N. A952), "Statistical theory of equations of state and phase transitions II. Lattice gas and Ising model," Phys. Rev., 87, 410-419. Leighton, W. and Scott. W. T. A939), "A general continued fraction expansion," Bull. Am. Math. Soc, 45, 596-605. Levin, D. A973), "Development of non-linear transformations for improving convergence of sequences," Int. J. Сотр. Math., B3, 371-388. Levin, D. A976), "General order Pade type rational approximants defined from double power series," J. hist. Math. Appl., 18, 1-8. Loeffel, J. J., Martiq, A., Simon. В., and Wightman, A, S., Phys, Lett., ЗОВ, 656-658.
Литература 475 Longman. I. M. A966), "The application of rational approximations to the solution of problems in theoretical seismology," Bull. Seismol. Sot: America, 56, 1045-1065. Longman. I. M. A972), "Numerical Laplace transform inversion of a function arising in viscoelasticity," Л Сотр. Phys., 10, 224-231. Longman I. M. A973), "Use of Pade table for approximate Laplace transform inversion," in P. R. Graves-Morris (ed.), Pade Approximants and Their Applications, Academic Press, London, pp. 13 U134. Longman. I. M. A974), "Best rational function approximation for Laplace transform inversion," SIAMJ. Math. Anal, 5, 574-580. Lopez. С and Yndurain. F. J. A973), "Model independent calculation of Kp dispersion relations; extrapolation with Pade techniques," Nucl. Phys. B, 64, 315-333. Lovelace. С A964), "Practical theory of three-particle states I Non-relativistic," Phys. Rev., 135B, 1225-1249. Lovelace, C. and Masson. D. A962), "Calculation of Regge poles by continued fractions," Nuovo Cim., 26, 472-484. Lubinsky. D. S. A980), "Exceptional Sets of Pad6 approximants," Ph.D. thesis, Univ. of Witwatersrand. Lubkin, S. A952), "A method of summing infinite series," /. Res. Nat. Bur. Stand., 48, 228-254. Luke, Y. L. A958), "The Pade table and the т-method," J. Math and Phys., 37,110-127. Luke, Y. L. A960), "On the economic representations of transcendental functions," /. Math, and Phys., 38, 279-294. Luke Y. L. A962), "On the approximate inversion of some Laplace transforms," in Proc. 4th. U.S. Nat. Cong. Appl. Mech., pp. 269-276. Luke. Y. L. A970), "Evaluation of the gamma function by means of Pade approximations," SIAMJ. Num. Anal., 1, 266-281. Luke, Y. L. A975), "On the error in the Pade approximants for a form of incomplete gamma function including the exponential," SIAMJ. Math. Anal., 6, 829-839. Luke. Y. L. A977), "On the error in Pade approximations for functions defined by Stieltjes integrals," Сотр. and Math, with Appl., 3, 307-314. Luke. Y. L. A978), "Error estimation in numerical inversion of Laplace transform using Pade approximation,"/. Franklin Inst., 305, 259-273. Luke. Y. L. A980), "Computations of coefficients in the polynomials of Pade approximations by solving systems of linear equations,"/. Сотр. and Appl. Math., 6, 213-218. Luke, Y. L., Fair, W., and Wimp, J. A975), "Predictor-corrector formulas based on rational , interpolants," Int. /. Computers and Math, with Appl., 1, 3-12. Lutterodt С L. A974), "A two-dimensional analogue of Pade approximant theory,"/. Phys. A, 1, 1027-1037. Lyness. J. and"Ninham, B. W. A967), "Numerical quadrature and asymptotic expansions," Math. Сотр., 21, 162-178. McCabe. J. H. A974), "A continued fraction expansion, with a truncation estimate, for Dawson's integral," Math. Сотр., 28, 811-816. McCabe. J. H. A975), "A formal extension of the Pade table to include two point Pade quotients,"/. Inst. Math. Appl., 15, 363-372. McCabe. J. 1Г. and Murphy J. Л. A976), "Continued fractions which correspond to power series at two points." /. Inst. Math. Appl., 17, 233-247. McCleod. J. B. A971). "A note on the e-algorithm," Computing, 7, 17-24.
476 Литература McCoy В. М. and Wu, Т. Т. A973), "The two dimensional Ising Model," Harvard Univ. Press, Cambridge. McEliece.R. J. and Shearer J. B. A978), "A property of Euclid's algorithm and its application to Pade approximation," SUM J. Appl. Math., 34, 611-615. Maehly. H. J. A956), October monthly progress report, Institute for advanced study, Princeton. Maehly. H. J. A960), "Rational approximations for transcendental .functions," in Proc. Int. Conf. on Inform. Proc, 1959, Butterworth, pp. 57-62. Maehly, H. J. and Witzgall. Ch. (I960), "Tschebyscheff— Approximationeninklcincn Interval- len II. Stetigkeitsatze fur gebrochen rationale Approximationen," Num. Math., 2, 293-307. Magnen, J. and Senior R. A976), "The infinite Volume limit of file $ model," Ann ¦ d 'hut. H. PoincareA, 24, 95-159. Magnus, A. A962), "Expansion of power series into P-functions," Math, 7.., 80, 209-216. Markov, A. A884), "Demonstration de certaines inegalitfc de Tchebychef," Math. Aim., 24, 172-180. Marx. I. A963), "Remark concerning a non-linear sequence to sequence transformation," /. Math, and Phys., 42, 334-335. Mason. J. C. A967), "Chebychev polynomial approximations for the i-membrane eigenvalue problem," SUM J. Appl Math., 15, 172-181. Masson. D. A967a), "Pade approximant and the partial-wave integral equation," J. Math. Phys., 8, 512-514. Masson. D. A967b), "Analyricity in the potential strength," J. Math. Phys., 8, 2308-2314. Mcinguet. J. A970), "On the solubility of the Cauchy interpolation problem," in Approximation Theory, ed. A. Talbot, Academic Press, London, 137-163. Mery. P. A977), "A variational approach to operator and matrix Pade approximation. Applications to potential scattering and field theory," in Pade and Rational Approxima- Approximation, eds. E B. Saff and R. H. Varga, Academic Press, New York, 375-388. Merz, G. A968), 'Tadesche Naherungsbruche und Iterationsverfahren hohcrer Ordnung," Сотр., 3,165-183. Michalik, B. A970), "Distortion operator method and Pade approximants," Лнн. Phys., 57, 201-213. Mills. W. H. A975), "Continued fraction and linear recurrences," Math. Сотр., 29, 173-180. Milne-Thomson L. M. A960), Calculus of Finite Differences, Macmfflan, London. de Montessus de Ballore, R. A902), "Sur les fractions continues algebriques," Bull. Soc. Math, de France, 30,28-36. _ dc Montessus de Ballore. R. A905), "Sur les fractions continues algebriques," Rend, di Palermo, 19,1-73. Muir. T. A960), A Treatise on the Theory of Determinants, Dover, New York. Murphy. J. A. A971); "Certain rational fraction approximations to (\+x2)~l/2," J. Inst. Math. Appl.,1, 138-150. Murphy. J. A. and O'Donohoe, M. R. A977), "A class of algorithms for rational approximation of functions formally defined by power series," /. Appl. Math, and Phys. (ZAMP), 28, 1121-1131. Narcowich. F. J. and Allen. G. D. A975), "Convergence of diagonal operator-valued Pade approximants to Dyson expansion," Comm. Math. Phys., 45, 153-157.
Литература 477 Nelson, E. A973), "Construction of quantum fields from Markoff fields," /. Fund. Anal, 12, 97-112. Newton.; R. G. A966), Scattering Theory of Wanes and Particles, McGraw-Hill. Ninham. B, W. and Lyness, J. A969), "Asymptotic expansions for the error functional," Math. Сотр., 23, 71-83. Nersett S. P. A974), "One step methods of Hermite type for numerical integration of stiff systems," BIT, 14, 63-77. Nuttall, J. A966), "Padfc approximants and bounds on the Bethe-Salpeter amplitude," Phys. Lett., 23, 492. Nuttall. J. A967), "Convergence of Pade approximants for the Bethe-Salpeter amplitude," Phys. Rev., 157, 1312-1316. Nuttall.- J, A970a), "Connection of Pade approximants with stationary variational principles and the convergence of certain Pade approximants," in The Fade Approximant in Theoretical Physics, eds. G. A, Baker, Jr. and J. L. Gammel, Academic Press, New York 219-230. Nuttall: J. A970b), "Convergence of Pad? approximants of meromorphic functions,"/. Math. Anal. Appl.,-31, 147-153. Nuttall, J. A973), "Variational principles and Pade approximants," in P. R. Graves-Morris (ed.), Pade Approximants and Their Applications, Academic Press, New York, pp. 29-40. Nuttall. J. A977), "The convergence of Pade approximants to functions with branch points," in E. B. Saff and R. H. Varga (eds.), Pade and Rational Approximation, Academic Press, New York, pp. 101-109. Nuttall. J. and Singh, S. R. A977), "Orthogonal polynomials and Pade approximants associated with a system of arcs," /. Approx. Theory, 21, 1-42. Ostrowski A. A925), "Uber Folgen analytischer Funktionen und etnig» Yerscharfungen des Picarden Satzes," Math. Zeit., 24,215-258. Pade, H. A892), "Sur la representation approchee d'une fonction par des fractions rationelles," Ann. de t'Ecole Normale Sup. 3tim' Sine, 9, Suppl., 3-93. Pade. H. A894), "Sur les series entieres convergents on divergent», et les fractions continues rationelles," Ada Math., 18, 97-112. Pade\ H. A899), "Memoire sur les developpements en fractions continues de la fonction exponentielle pouvant servir d'introduction a la theorie des fractions continues algebriques," Ann. Sci. Ecole Norm. Sup., 16, 395-426. Pade, H. A900), "Sur la distribution des reduites anormales d'une fonction," Comptes Rendus, 130, 102-104. Pade. H. A901), "Sur 1'expression generate de la fonction rationelle approchee de A +x)m" Comptes Rendus, 132, 754-756. Pade, H. A907), "Recherches sur la convergence de developpements en fractions continues d'une certaine categorie dc fonctions," Ann. Sci. Ecole Norm. Sup,, 24. Paydon. 3. F. and Wall, H. S. A942), "The continued fraction as a sequence of linear transformations," Duke Math. /., 9, 360-372. Pekeris, С L. A958), "Ground state of two electron atoms," Phys. Reo., 112, 1649-1658. Pekeris. С L. A959), "Binding energy of Helium," Phys. Rev., 115, 1216-1221. Peres A. A963), "Mechanical model for quantum field theory,"/. Math. Phys., 4, 332-333. Perron. O. A957), Die Uhrevon den KettmbrMen, B. G. Teubner, Stuttgart.
478 -Литература Pfeuty. P., Jasnow. D., and Fisher M. E. A974), "Cross over scaling functions for exchange anisotropy," Phys. Rev. B, 10, 2088-2112. Pindor. M. A976), "A simplified algorithm for calculating the Pade table derived from the Baker and Longman schemes," J. Сотр. Appl. Math., 2, 255-258. Pindor M. A979a), "Pade approximants and rational functions as tools for finding poles and zeros of analytical functions measured experimentally," in L. Wuytack (ed.), Pade Approximation and Its Applications, Academic Press, New York, pp. 338-347. Pindor M. A979b), "Matrix variational Pade approximants and multistep square well poten- potentials," Lett. Math. Phys., 3, 223-228. Pindor. M. A980), "Unambiguous results from variational matrix Pade approximants," CERN report. Pommerenke. Ch. A973), "pade approximants and Convergence in capacity," Л Math. Ana!. Appl,, 41, 775-780. Pringsheim. A. A910), "Uber Konvergenz und Funktionen-theoretischen Charakter gewisscr limitarperiodischer KettenbrBche," Sitzungsberichte der Math.-Physikalische Klasse der tCgt. Sayerischen Akadetnie der Wissenschaften zu Mimchen, 6, 1-52. Reed M. and Simon, B. A979); "Scattering Theory," vol. 3 in Methods of Modern Math. Physics, Academic Press, New York. Rehr. J. J., Joyce, G. S., and Guttmann, A. J. A980), "A recurrence technique for confluent singularity analysis of power series," J. Phys. A, 13, 1587-1602. Reid, W. T. A972), Rkcati equations, Academic Press, London. Richardson L. F. A927), "The deferred approach to the limit, I. Single lattice," Phil, Trans. Roy. Soc. Land., A226, 299-349. Riesz; F. and Nagy, B. v. Sz. A955), Functional Analysis, Ungar. Riesz. M.'A922a), "Sur le probleme des moments," Arkiv for Matematik, Astronomi vch 'Fysik, 16 A2), 1-23. Riesz, M. A922b)i "Sut le probleme des moments," Arkiv for Matematik, Astronomi och Fysik, 16A9), 1-21. Riesz M. A923), "Sur le probleme des moments," Arkiv for Matematik, Astronomi och Fysik, 17,A6), 1-52. Riordan I A966), Combinatorial Identities, Щку. Rissanen. J. A973), "Algorithms for triangular decomposition of block Hankel and Toeplitz matrices with applications to factoring positive matrix polynomials," Math. Сотр., 27, 147-155. Rivlin T. J. A969), Ail Introduction to the Theory of Functions, Blaisdell. Roberts. D. E. A977), "An analysis of double power series using rotationally covariant approximants,"Л Сотр. Appl. Math.,3, 257-262. Roberts. D. E., Griffiths. H. P., and Wood. D. W. A975), "The analysis Of double power Series using Canterbury approximants,"/. Phys. A, 8, 1365-1372. Robinson P. D. and Barnsley M. F. A979), "Pointwise bivariational bounds on.Solutions of Fredholrh integral equations," SIAMJ. Num. Anal., 16, 135-144. Regers. I., i. A907), "On the representation of certain asymptotic series as convergent continued fractions," Proc. Loud. Math. Soc., 4, 72-89. Roth. A. A938), "Approximationseigenschaften und Strahlengrenzwerte unendlich vieler lin- earer Gleichungen," Comm. Math. Helv., 11. 77-125. Roth R. A»65), "The qualifying examination," Math. Mag., 38, 166-167.
Литература 479 Rudin, W. A976), "Principles of Mathematical Analysis," McGraw-Hill. Ruijgrok Th. W. A968), "A new formulation of the theory of strong coupling," Nucl. Phys. B, 8,591-608. Ruijgrok.. Th. W. A972), "A numerical study of analytieity In the coupling constant," Nucl. Phys. В. 39, 616-642. Runge. С A885), "Zur Theorie der eindeutlgen analytischen Funktionen," Ada Math,, 6, 229-244. Rulishauser. H. A954), "Der Quotienlen-Differenzen-AIgorithmus," 7.eli. fur Angewandie Math., 5, 233-251. Rutishauser. H. A957), Per Quotienten-Differenzen-AtgorUhmus, Birkhauser Verlag, Basel. Saff, E. B. A972), "An extension of Montessus de Ballore's theorem on the convergence of interpolating rational functions,"/. Approx. Theory, 6, 63-67. Saff. E. В., Schonhage. A., and Varga R. S. A978), "Geometrical convergence to e~: by rational functions with real poles", Num. Math., 25, 307-322. Saff. E. B. and Varga. R. S. A975), "On the zeros and poles of Pade approximants to r," Num. Math. ,25, 1-14. Saff E. B. and Varga R S. A976a), "Zero-free parabolic regions for sequences of polynomials," SIAMJ.Math. .ля/., 7,-344-357. Saff, E. B. and Varga. R. S. A976b), "On the sharpness of theorems concerning zero free regions for certain sequences of polynomials," Num. Math., 26, 345-354. Saff, E. B. and Varga R. S. A977), "On the zeros and poles of Pade approximants to e". II," itt E. B. Saff and R. S. Varga (eds.), Pade and Rational Approximation, Academic Press, New York, pp. 195-214. Saff, B. B. and Varga. R. S. A978), "Non-uniqueness of best rational approximations to real functions on real intervals,"/, Approx. Theory, 23, 78-85. Saff, E В., Varga. R. S-, and Ni, W.-C. A976), "Geometric convergence of rational approxima- approximations to e~" in infinite sectors," Num. Math., 26, 211-225. Saber H. E. A962), "Note on osculatory rational interpolation," Math. Сотр., 16, 486-491. Schlessinger L. and Schwartz. C. A966), "Analytieity as a computational tool," Phvs. Rev. Lett., 16, 1173-1174. Schoenberg I. J. A951), "On the Polya frequency functions I: The totally positive functions and their Laplace transforms,"/. d'Anal. Math., 1/331-374. Schwartz. С A966), "Information content of the Born series,"/. Сотр. Phys., 1, 21-28. Schwartz. C. and Zemach. С A966), "Theory and calculation with the Bethe-Salpeter equation," Phys. Rev., 141, 1454-1467. Schweber, S. S. A961), Introduction to Relativistic Quantum Field .Theory, Harper and Row, New York. Schofield. D. F. A972), "Continued fraction method for perturbation Гпеогу," Phvs. Rev. Lett., 29,811-814. Scott- W. T. and Wall H. S. A940a), "Continued fraction expansions for arbitrary power series," Ann. Math., 41, 328-349. Scott W. T. and Wall. H. S. A940b), "A convergence theorem for continued fractions," Trans. Am. Math. Soc, 47,155-172. Shafer R.E, A974), "On quadratic approximation," SIAM J. Sum. Anal:, 11,447-460. Shanks D. A955), "Non-linear transformations of divergent and slowly convergent sequences," J. Math. Phys,, 34, 1-42.
480 Литература Sheludyak. Yu. E. and Rabinovich. V. A. A979), "On the values of the critical indices of the_ three dimensional Ising model," High Temp., 17,40-43. Short. L. A978), "The practical evaluation of multivariate approximants; with branch points," Proc. Roy. Soc. London A, 362, 57-69. Sidi. A. A979), "Convergence properties of some non-linear sequence transformations," Math. Сотр., 33,315-326. Sidi, A. A980a), "Some aspects of two-point Pade approximants,"/. Сотр. Applet* Math., 6, 9-17. Sidi. A. A980b), "Analysis of the Convergence of the T-lransformation for power series," Math. Сотр., 35, 833-850. Siemieniuch. I. L. A976), "Properties of certain rational approximations to e"*,'1 S.I.T., 16, 172-191. Simon. B. A970), "Coupling constant analyticity for the anharmonic oscillator," Ann. Phys., 58, 76-136. Simon, B. A972), "The anharmonic oscillator—a singular perturbation theory," in D. Bessis (ed.), Сагфе Lecture Notes in Physics, Vol. 5, Gordon and Breach, pp. 383-414. Smith. D. A. and Ford W. F. A979), "Acceleration of linear and logarithmic convergence," SIAMJ. Num. Anal., 16, 223-240. Smith, I. M., Siemieniuch. J. L., and Gladwell I. A977), "Evaluating Nersett methods for integrating differential equations in time," Int. J. Num. Anal. Meth. in Geomechanics, 1, 57-74. Smithies. F, A958), Integral Equations, Cambridge University Press. Starkand, Y. A976), "Subroutine for calculation of matrix Pade approximants," Comtn. Сотр. Phys., 11, 325-330. Stern, M. S. and Warburton A. E. A. A972), "Finite difference solution of the partial wave Schr6dinger equation,"/. Phys. A, 5, 112-124. Stieltjes. T. J. A884), "Quelques recherches sur les quadratures dites mechaniques.'Mnn. Sci. Ecole. Norm. Sup., 1, 409-426. Stieltjes, T. J. A889), "Sur la reduction en fraction continue d'une serie precedent suivant les puissances descendents d'une variable," Ann. Fac. Set. Toulouse, 3H, 1-17... Stieltjes, T. J. A894), "Recherches sut les fractions continues," Ann. Fac. Sci. Toulouse, 8J, 1-122; 9A, 1-47. Stoer J. A961), "Uber zwei Algorithmen zur Interpolation mit rationalen Funktlonen," Num. Math., 3,285-304. Stone. M. H. A932), Linear Transformations in Hilbert Space, Am. Math. Soc, Providence. Takehasi and Mori, A971), "Estimation of errors in the numerical quadrature of analytic functions," Applicable Analysis, 1, 201-229. Talbot. A. A979), "The accurate numerical inversion of Laplace transforms,"/. Inst. Math. Appl., 23, 97-120. •Tani. S. A965), "Padc approximants in potential scattering," Phys. Reo., 139, B1011-1020. ¦Tani. S. A966a), "Complete continuity of the kernel in generalized potential scattering I. Short range interaction without strong singularity," Ann. Phys., 37, 411-450. Tani. S. A966b), "Complete continuity of the kernel in generalized potential scattering II, Generalized Fourier series expansion," Ann. Phys., 37, 451-486. Taylor J. M. A978), "The condition of Gram matrices and related problems," Proc. Hoy, Soc. Mn., 80A, 45-56.
Литература 481 Tebebycheff P. A858), "Sur les fractions continues," /. de Math., 8, 289-323. Tchebycheff. P. A874), "Sur les valeurs limites des integrales,"/. de Math, Pures et'Appl, 19, 157-160. Thacher H. С and- Tukey. J. A960), "Rational interpolation made easy by a recursive algorithm," unpublished manuscript. Thacher, H. С A974), "Numerical application of the generalized Euler transformation," in J. L. Rosenfeld (ed.), Information Processing 74, North Holland, Amsterdam, pp. 627-631. Thiele, T. N. A909), "Interpolationsrechnung," Teubner. Thompson. С J., Guttman.-A. J., and Ninham. B. W. A969), "Determination of critical behaviour in lattice statistics from series expansions, II," J. Phys. C, 2, 1889-1899. Thron W. J. A948), "Some properties of continued fractions 1+482 Литература Van Vleck, E. В. A904), "On the convergence of algebraic continued fractions whose coeffi-. cicnts. have limiting values," Trans. Am. Math. Soc, 5, 253-262. Var&a. R. S. A961), "On higher order stable implicit methods for solving parabolic partial differential equations," J. Math. Phys., 40, 220-231. Varga R. S. A962), Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs. Villani. M. A972), "A summation method for perturbative series with divergent terms," in D. Bessis (ed.), Cargese Lecture Notes in Physics, Vol. 5, Gordon and Breach, New York, pp. 461-474. Viskovatov B. A803-1806), "De la methodc generate pour reduire toutcs Sortes des quantites en fractions continues," Memoires tie L'Academie Imperiale des Sciences de St, Peters- Petersburg, 1, 226-247. Yorobyev. Yu. V. A965), Method of Moments in Applied Mathematics, Gordon and Breach, New York. Waadeland H. A979), "General Г-fractions corresponding to functions satisfying certaia boundedness. conditions,"/. Approx. Theory, 26, 317-328. Wall, H. S. A931), "On the Padc approximants associated with a positive definite power series," Trans. Am. Math. Soc, 33, 511-532. Wall. H. S. A932a), "On the relationship among the diagonal files of a Fade table," Bull, Am. Math. Soc, 38, 752-760. Wall. H. S. A932b), "General theorems on the convergence of sequences of Padc approxi- approximants," Trans. Am. Math. Soc, 34, 409-416. Wall. H. S. A945), "Note on a ccrtairfcontinued fraction," BulL Am. Math, Soc, 51,93Q-934. Wall. H. S. A948), The Analytic Theory of Continued Fractions, Van Nostrand, Princeton, NJ, Wallin H. A972), "On the convergence theory of Pade approximants," in P, L. Butzer, J.-P. Kahane, and B. Sz-Nagy (eds.), Linear Operators and Approximation XProc. Conf. at Oberwolfacbl, Birkhiiuser, Basel, pp. 461-469. Wallin H. A974), "The convergence of Pade approximants and the size of the power scries coefficients," Appl. Anal., 4,235-251. Walsh. J. L. A964a), "Convergence of sequences of rational functions of best approximation, I," Math. Ann., 155,252-264. Walsh J. L, A964b), "Pade approximants as limits of rational functions of best approximation," J. Math. undMech., 13, 305-312. Walsh, J. L. A965a), "Convergence of sequences of rational functions of best approximation II," Trans. Am. Math. Soc, 116, 227-237. Wakh J. L. A965b), "The convergence of sequences of rational functions of best approxima- approximation with some free poles," in Approximation of Functions, ed. H. L, Garabedian, Elscvier, Amsterdam, 1-16. Walsh, J. L. A967), "On the convergence of sequences of rational functions" S1AMJ. Num. Anal., 4,211-221. Walsh J. L, 41969), "Interpolation and approximation by rational functions in the complex domain," Am. Math. Soc. Colloq, Pub., Vol. 20, Providence, R.I. Walsh T. L. A970), "Approximation by rational functions; open problems,1 V. Approx, Theory, 3,236-242. Warburton, A. E A. and Stern M. S. A968), "Two body off-shell potential scattering," Kuoco On;., 60A, 131-159. D. D. A976), "An extension of Saff.s theorem on the, сопу«?Ш;« Литература 483 Watson P. J. S. A973). "Algorithms for differentiation and integration." in P. R. Graves-Morris (ed.), /We Approximates and Their Applications. Academic Press. London, pp. 93-98. Watson P. J. s. A974). "Two variable rational opproximants-a new method,"/. Phys. A, 7, L167-L170. Werner. H, A979), "A reliable method for rational interpolation," in L. Wuytaek (ed.). Pads Approximation and Its Applications, Springer Lecture Notes in Mathematics, No. 765, pp. 257-277. Werner. H. A980),"Ein algorithmic zurrationalen Interpolation," in Numerischc Methoden der Approximationstheorie, 5, ed& L. Collatz, G. Meinardus and H. Werner, Birichiuuscr, 319-337. Werner H. and Schabaek R. A972), "Praktische Mathcmatik H," Springer Vcrlag, Berlin. Werner. H. and Wnytack L. A978), "Non-linear quadrature rules in the presence of a singularity," Сотр. & Math, with Appkiis.,4,237-245 A981). Wetlerling W. A963), "Ein Interpolationsverfahrcn zur Losung dcr linearcn Glcichungssys- tcme, die bci der rationalen Tchcbyschcf f-Approximation auf treten," Archto. Rat. Medt. undAmil., 12,403-408. Wheeler, J. C. and Gordon. R. G. A970), "Bounds for averages using moment constraints," in G. A. Baker. Jr., and I L. Gammel (eds.), The Fade Approximant in Theoretical Physics, Academic Press, New York, pp. 99-128. Widde'r D. V. A972), The Laplace Transform, Princeton University Press. Wilkinson J. H. A963), Rounding Error in Algebraic Processes, Notes in Applied Science, No. 32, H.M.S.O., London. .Wilkinson. Т. Н. A965), The Algebraic Eigenvalue Problem, Oxford University Press. Wilson. R. A927), "Divergent continued fractions and polar singularities," Proc. Loud. Math. Soc.,26, 159-168. Wilson R. A928a), "Divergent continued fractions and polar singularities II. Boundary pole multiple," Proc. bond. Math. Soc, 27,497-512. Wilson. R. A928b), "Divergent continued fractions and polar singularities Ш, Several boundary poles," Proc. Land. Math. Soc, 28,128-144. Wilson R. A930), "Divergent continued fractions and non-polar singularities," Proc. Lond. Math. Soc., 30,38-57. ¦ Wood T>. W. and Griffiths. H. P. A974), "Cbisholm approxunants and critical phenomena," J. ^l104 Wright. K, A970), "Some relationships between implicit Runge-Kutta, collocation and Lanczos т-methods and their stability properties," BIT, 10,217-227. Wuyfack L. A973), "An algorithm for rational interpolation similar to the q.d.-algorithm," Num. Math., 20 418-424. Wuylack. L. A974a), "Numerical integration by using non-linear techniques"/. Сотр. Appi. Math., 1,261-212. Wuylack L. A974b), "On some aspects of the rational interpolation problem," SIAMJ. Num. Anal., 11, 52-60. Wuyfack L. A975), "On the osculatory rational interpolation problem," Math. Сотр., 29, 837-843. Wuytaek L. A979), "Commented bibliography," in L. Wuytaek (ed.), Pa'de Approximation and Us Applications, Springer Lecture Notes in Mathematics, No. 765, pp. 375-392. Wynn P. A956), "On a device for calculating the em{Sn) iransformation," Math. Tables and A.C., 10, 91-96.
484 Литература Wynn P. A960), "The rational approximation of functions which are formally defined by power series expansions," Math. Сотр., 14,147-186. Wynn, P. A961a). "The cpsilon al|orithm and operational formulas of numerical analysis," Math. Сотр., 15,151-158. Wynn, P. A961b), "LValgoritmo e la tavola di Pade," RentS. Mat. Roma, 20,403-408, Wynn. P. A962a), "The numerical efficiency of certain continued fraction expansions," Proc. Kon.Ned.Akad. Weiensch. A,65,127-154. Wynn P. A962b), "Acceleration techniques for iterated vector and matrix problems," Math. Сотр., 16,301-322. Wynn, P. A963), "Continued fractions whose coefficients obey a non-commutative law of multiplication," Ardt. Rat. Mech. Ana!., 12,273-312. Wynn. P. A964), "General purpose vector epsilon algorithm procedures." Num. Math., 6, 22-36. Wynn. P. A966a), "Upon systems of recursions which obtain among quotients of the Pad& table," Num. Math., 8,264-269. Wynn P. A966b), "On the convergence and stability of the E-algorithm," SIAMJ. Num. Anal, 3^91-122. Wynn. P. A967), "A general system of orthogonal polynomials," Quart. J. Math., 18,81-96. Wynn, P. A968), "Upon the Pade table derived from a Stieltjcsseries." SIAMJ. Num.Anal» 5, 805-834. Wynn. P. A973), "On the zeros of certain confluent hypcrgeometric functions," Proc. Am. Math. Soc, 40; 173-182. Wynn P. A974), "Some recent developments in the theories of continued fractions and the Pade table," Rocky Mm. J. Math., 4,297-323. Wyttfl. P. A977), "The transformation of series by the use of Pade quotients and more general approximants," in E. B. Saff and R. S. Varga (eds.), Pade and Rational Approximation, Aeademic Press, New York, pp. 121-146. Young R. C, Biedenharn, L. C, and Feenberg, E. A957), "Continued fraction approximants to the Brillouin-Wigner Perturbation series," Phys. Reo., 106.1151-1155. Zinn-Justin. J. A970), "Strong interaction dynamics with Pade approximants," /Уда. Repts., 1С, 56-102. Zinn-Justin J. A971), "Convergence of Pade approximants in the general case," in A, Visconti (ed.), Colloquium on Advanced Computing Methods in Theoretical Physics, CNRS, . Marseille. Zinn-Justin. J. A973), "Recent developments in the theory of Pade approximants," in A. Visconti (ed.), International Colloquium on Advanced Computing Methods in Theoretical Physics, Vol. 2, CNRS, Marseille, p. С-ХШ-1. Zohar s. A974), "The solution of a Toeplitz set of linear equations,"/. Assoc. Сотр. Meclu. 21,272-276.
Основные обзоры Baker. G. A., Jr. A965), "The theory and application of the Pade approximant method," Advances in Theoretical Physics, 1,1-58. Baker. G. Л., Jr. A975), The Essentials of Pade Approximants, Academic Press, New York. Baker G. A., Jr. and Gammel, J. L,, eds. A970), The Padi Approximant in Theoretical Physics, Academic Press, London. Basdevant. J.-L. A972), "The Pade approximate and its physical applications," Fortschr. Phys.% 20,282-331. Bausset M., ed. A973), Acceleration des Convergences, Centre Universitaire de Toulon. Bessis D., ed. A972), Carg'ese Lectures in Physics, Vol. 5, Gordon and Breach, New York. Bessis, D., Gilewicz. J., and Mery P., eds. A975), Proceedings of a Workshop on Pade Approximates, C.N.R.S., Marseille. Brezinski, C. A977), Acceleration de la Convergence en Analyse Numerique, Springer Lecture Notes in Mathematics, No. 584. Brezinski, С A980), "Pade type approximation and general orthogonal polynomials," Birkhauser Verlag. Cabanncs H., ed. A976), Pade Approximant Method and its Application to Mechanics (Proceed- (Proceedings of the Euromcch Colloquium, Toulon, 1975), Springer Lecture Notes in Physics, No. 47. Chui C. K. A976), "Recent results on Pade approximants and related problems," in G. G. Lorentz, C. K. Chui and L. L. Schumaker (eds.), Approximation Theory, Academic Press, New York, pp. 79-116. Claessens. G. A976), "Some aspects of thft rational Hermite interpolation table and its applications," Ph. D. Thesis, University of Antwerp. Gilewicz. J. A978), Approximants de Pade, Springer Lecture Notes in Mathematics, No. 667. Gragg, W. B. A972), "The Pade table and its relation to certain algorithms of numerical analysis," SI AM Review, 14, 1-62. Graves-Morris, P. R., ed. A973), Pade Approximants and Their Applications, Academic Press, London. Graves-Morris. P. R., ed, A973), Pade Approximants, Inst. of Phys. Press, London. Jones. W. B. and Thron. W. J. A974), "Proceedings of the international conference on Pade approximants, continued fractions and related topics," Rocky Mtn.J. Math., 4,135-397. Pade. H. A892), "Sur la representation approchee d'une fonction par des fractions rationelles," Annales de I'Ecole Normale, 9, suppl. 3, 93. Saff, E B. and Varg,a. R, S., eds. A977), Pade and Rational Interpolation, Academic Press, New York. Warner D. D. A974), "Hermite interpolation with rational functions," Ph. D. Thesis, Univer- University of California. Wimp. J. A981), "Sequence Transformations and Their Applications," Academic Press, New York. Wuytack L. A976), "Applications of Pade approximation in numerical analysis," in Schaback, R. and Scherer, K. (eds.), Approximation Theory, Springer Lecture Notes in Mathe* matics, Vol. 556, pp. 453-461. Wuytack L., ed. A979), Pade Approximation and Its Applications, Springer Lecture Notes in Mathematics, Vol. 765. Zinn-Justin. J. A970), "Strong interaction dynamics with Pade approximants." Phps. Reports, 1С, 56-102.
Избранная библиография о непрерывных дробях и методе моментов Akhiczcr. N. I. A965), The Classical Moment Problem, Oliver and Boyd, London. Jones W. B. and Thron, W. J. A980), "Continued fractions: Analytic theory and applications," in G.-C. Rota (ed.). Encyclopedia of Mathematics and Its Applications, Addison-Wesley, Reading,, Mass. Khovanskii A. N. A963), Application of Continued Fractions and Their Generalizations to Problems in Approximation Theory, Noordhoff. Perron 0. A957), Die Lchre von det Keltenbrikhen, Vol. 2, Teubncr B. G., Stuttgart. Sholiat J. A. and Tamarkin J. D. A963), The Problem of Moments, American Math. Sou Publ.. Vol. 1, Providence, R.I. Vorobyev. Yu. V. A965), Method of Moments in Applied Mathematics, Gordon and Breach, New York. Wall. H. S. A929), "On the Pade approximants associated with the continued fraction and series of Stieltjes," Trans. Am. Math. Soc, 31, 91-115. Wall H. S. A948), The Analytic Theory of Continued Fractions, Van Nostrand, Princeton. Общая литература 1) Basdevant, J.-L. and Lee, B. W. A969a), "Fade approximants in the Литература 487 Baulioff. W. A977), "Padi approximants in the Wiek-Cutkosky model," J. Phys. A, 10, 129-134. Beardon, A. F. A968a), "The convergence of Padc approximants," J. Math. Anal. Appi., 21, 344-346. Bcardon, A. F. A968b), "On the location of poles of Padc approximants,"./. Math. Ana!. Appl., 21. 469-474. Bell, (i. I. and (ilasstonc, S. A970), Nuclear Reactor Tltcor'v, van Nostrand-Reinhold, New York. Bender. C. and Wu, Т. Т. A968). "Analytic structure of energy levels in a field theory model," Phys. Rec. Lett.. 21, 406-409. Benofy. L. P. and Gamniel. J. L. A977), "Variational principles and matrix Pade approxima\ in E. B. Saff and R. H. Varga^cd.), Pade and Rational Approximation, Academic Press, New York. 339-356. Benofy, L. P., Gammcl, J. L, and Mery. P. A976), "The off-shell momentum as a variational parameter in calculations of matrix Padc approximants in potential scattering," Phys. Ла-.Л. 13, 3111-3114. Bernstein. S. A928), "Sur les functions absolument monotones," Acta Math., 52, 1-66. Besisis. D., cd. A972), Cargisc Lectures in Physics, Vol. 5, CJordon and Breach, New York. Bessis, D. A973), P. R. Graves-Morris (cd.). Topics in the Theory ofPatli' Approximants, Inst. of Phys., Bristol. 19-44. Bessis, D. A976). "Construction of variational bounds for the A'-body cigenstate problem by the method of Pade approximants," in Pade Approximants Method and its Application to Mechanics, H. Cabanncs (ed.). Springer Lecture Notes in Phys., No. 37. Berlin, 17-31. Bessis. D. A979), "A new method for the combinatorics of the topological expansion," Contm. Math. Phys.,69, 147-163. Bessis. D.. Drouffe. J. M.i and Moussa, P. A976), "Positivity constraints for the Ising ferromagnetic model."./. Phys. A, 9, 2105-2124. Bessis, D., Epele, L., and Viltani. M. A974), "Summation of regularized perturbation cxpan- sions for singular interactions."./. Math. Phys., 15, 2071-2078. Bessis, D., Mery. P., and Turehetti, G. A974), "Angular momentum analysis of four nuclcon Green's function," Phys. Rev. P, 10. 1992-2009. Bessis. D., Mery, P.. and Turehetti, G. A977), "Variational bounds from matrix Padc approximants in potential scattering," Phys. Rev. D, 15. 2345-2353. Bessis, D.. Moussa, P., and Villani, M. A975). "Monotonie converging variational approxi- approximants to the functional integrals in quantum statistical mechanics,",/. Math. Phys., 16, 2318-2325. Bessis, ft. and Pusterla, M, A967), "Unitary Padc approximants for the S-matrix in strong coupling field theory and application (o the.calculation of p and/0 Reggc trajectories," Phys. Lett., 25B, 279-281. Nuttall. J. A480), "Sets of minimum capacity, Pade opproximanfs and the bubble problem," in С Runlos :md IX Bessis (oils.). Bifurcation Pmblenu in Mathematical Physics and Related Topics, I). Rcidd, Boston, pp. lW-Ж
Избранная литература по аппроксимациям Паде и критическим явлениям Baker, С5. A.. Jr. A967), "Convergent, bounding approximation procedures to the fcrromag. netic Ising model," Phys. Rev., 161,434-445. Baker, G. A., Jr. A973). "Generalized Padc approximate bounds for critical phenomena." in P. R. Graves-Morris (ed.), Padi Approximates and Their Applications, Academic Press. London, pp. 147-158. Baker, G. A., Jr. A975), "Self-interacung-boson quantum field theory.an.d the thermody'namic . limit in ddimensions,"/. Math. Phys., 16, 1324-1346. Baker, G. A., Jr. A977b), "Analysis of hyperscaling in the Ising model by the high temperature scries method," Phys. Rev. B, 15,1552-1559. Baker, O. A.. Jr. A979), "The continuous-spin Ising model of field theory and the renormaliza- tion group," in C. Bardos and D. Bessis (eds.). Bifurcation Phenomena in Mathematical Physics and Related Topics, D. Reidel, Boston. Baker, G. A.. Jr.. Gilbert, H. E., Eve, J., and Rushbrooke, G. S. A967), "High temperature expansion for the spjn-i Heisenberg model," Phys. Rev., 164, 800-817. Baker, G. A., Jr., Nickel, B. G., and Meiron. D. I. A978), "Critical indices from perturbation analysis of the Callan-Symanzik equation," Phys. Rev. B, 17, 1365-1374. Baker, G. A., Jr., Rushbrooke, G. S., and Gilbert, H. E. A964), "High temperature series expansions for the spin-, Heisenberg model by the method of irreducible representa- representations of the symmetric group," Phys. Rev.. 135, A1272-1277. Domany. E., Mukamcl, D.. and Fisher. M.E. A976). "Destruction of first order transitions by symmetry breaking fields." Pliys. Rev. B, 15, 5432т- 5441. Essam, J. W. and Fisher, M. E. A963). "Padi approximant studies of the lattice gas and Ising fcrromagnet below the critical point,'' J. Chem. Phys., 38, 802-812. Fisher, M. E A967), "The theory of equilibrium critical phenomena." Rept. Prog. Pins., 30. 615-731. Fisher. M. E. and Burford, R. J. A967), "Theory of critical point scattering and correlations I. The Ising model." Phys. Rev., 156, 583-622. Gaunt, D. S. and Guttmann. A. J. A974). "Asymptotic analysis of coefficients." in С Doinb and M. S. Green (eds,). Phase Transitions and Critical Phenomena, Vol. 3. Academic Press. New York. pp. 181-245. Oiuint, D. S. and Guttmann, A. J. A974), "Asymptotic analysis of coefficients," in C. Domb and M. S. Green (eds.). Phase Transitions and Critical Phenomena, Vol. 3, Academic Press. New York, pp. 181-245. Griffiths, R. В., "Rigorous results and theorems," in С Domb and M. S. Green (eds.)! Phase Transitions and Critical Phenomena, Vol. 1. Academic Press, London, pp. 7-109. Guttmann. A. J., Thompson, C. J.. and Ninham, B. W. A970), "Determination of critical' behaviour from series expansions in lattice statistics. IV." J. Pliys. С 3.1641-1651. Hunter. D. L. and Baker. G. A., Jr. A973), "Methods of series analysis I," Phys. Rec.B, 7, 3346-3376. Moore, M. A.. Jasnow. D., and Wortis. M. A969). "Spin-spin correlation function of the three-dimensional Ising ferromagnel above the Curie temperature," Phys. Rev. Lett., 22, 940-943. Nagle. J. F. and Bonner, J. C. A970), "Numerical study of Ising chain with long range ferromagnetic intcraclioBS.'V« Phys. C, 3,352-365, '
Избранная литература по аппроксимациям Паде и теории электрических цепей Brophy, F. and Salazar. А. С. A973), "Considerations of the Padc approximant technique in Ihe synthesis of recursive digital filters." IEEE Tram. Audio Electroaawt.. Au-21, 500-505. Chui, C. K... and Chan. A. K'.( 1974). "A two sided rational approximation method for recursive digital filtering." /./:'. /:'./;. Trans. Acous. Speech ami Signal Proc, ASSP-27 141-145. Cluii, C. K., Smith. P. W., and Su. L. Y. A977). "A minimization problem related to Pad6 synthesis of recursive digital filters." in E B. Saff and R. S. Varga (eds.). Puile and Rational Approximation, Academic Press, New York. pp. 247-256. Fahmy. M. F. A980). "The use of Padc approximates in the derivation of low-pass filters with simultaneous flat amplitude and delay characteristics." Circuit Tfwon: 8, 197-204. Hastings-James R. and Mchra. S. К. AЧ77). "Extensions of the Padc approximant technique '•¦tot the design of recursive digital filters." 1KI-'K Trans Assp., 501—509. Scanlan, I. O. A973), "Circuit theory," in P. R. Graves-Morris (ed.), Pcute Approximants, Institute of Physics, Bristol, pp. 101—111. Scott, R. E. A977). "Low order rational approximations to the delay operator." HIKE Tram: Ind. lllevtron. ami Control Iimmm.. IECI-24, 61-64. Shamash. Y. A975), "Linear system reduction using Padc approximation to allow retention of dominant modes." Int. J. of Control, 21, 257- 272. Sobhy. M. 1. A973). "Applications of Pade upproximants in electrical network problems." in V. R. Graves-Morris (cd.), Padc Approsumints and Their Applications. Academic Press. London, pp. 321-.% Литература по аппроксимациям Паде для специальных функций и стандартных машинных функций Икс. С. Т. A968), Computer Ifruhtotion of Mathematical Functions, Prcntke-НаП. Ciautsclii. W. A975). "Computational methods in special functions a survey," in R. A. Askcy (ed.). Theory mid Application of Special Functions, Academic Press, New York, pr». 1 98. Kogbctliaii7.,15. (}. (I960), "Generation of elementary functions," in A. Ralston and 11. S. Wilf {eds.),. Mathematical Methods for Digital Computers, Wiley, pp. 7-35. *Lukc. Y. L. A964), The Special Fumticms and Tlwir Approximations. Vol. I, Academic Press. New York. ¦ Lttkc. Y. L. A969), Tlie Special Functions and Their Approximations. Vol. II, Academic Press. New York. Machly. H. J. (I960). "Rational approximations for transcendental functions," in РгосееЛпцх of the International Conference on Information Processing, Hutterworth, London, pp.
Литература по аппроксимациям Эрмита — Паде Для заданного набора (по крайней мере двух) функций fa (х) с разложе- разложениями в степенные ряды 2?/. «=1. 2,.... п, i=o где //@) Ф 0, /=1, 2 п, задача Эрмита—Паде состоит в том, чтобы най- найти полиномы Qi (х) степени соответственно не выше <3{, такие, что 2 fcMQ/M=O(«°). п где а=п — 1 + 2 °i- см- § 1-3. (= 1 de Bruin. М. О. A974), "Generalized continued fractions and a multi-dimensional Padc'table* Ph. D. thesis, University of Amsterdam. <5c Bruin, M. G. A978), "Convergence of generalized C-fractions," J. Approx. Theory, 24, 177-204. Chudnovsky. G. V. A980), "Padc approximation and the Ricmann monodromy problem," in С Bardos and D. Bcssis (cds.). Bifurcation Problems in Mathematical Physics und Related Topics, D. Rcidcl, Boston, pp. 449-510. Delia Dora, J. and Di Crcsccnzo, C. A979), "Approximation de Padc-Hcrmite," in L. Wuytack (ed.), Padc Approximation and Its Applications, Springer Lecture Notes in Mathematics, No. 765, Berlin, pp. 88-115. Hermite, Ch., A917), "Sur quelques approximations alg6briques." Oeuvres; I. Ill, 146-149: "Sur la fonction-cxponenticllc," Oeuvres, t. Ill, 150-181; "Sur la generalisation des fractions continues algebriqucs," Oeuvres, t. IV, 357-377, Gainhicr-Yillars, Paris. Mahler, K. A968), "Perfect systems," Compositio Math., 19, 95-166. Padc, H. A894), "Sur lag&neralisation des fractions continues algcb'riqucs/V. de Math. Pun's et Appl., 10,291-329.
Литература по аппроксимациям Паде, квазианалитическим функциям и шумам Froissart, М. A969), private communication to J. L. Basdcvant, in "Pade approximants," in M. Nikolic (ed.), Meihodi ofSulmudear Physics, Gordon and Breach, pp. 1-60. Ciammcl, J. L. A973), "Effect of random errors (noise) in the terms of a power scries on the convergence of the Pade approximants," in P. R. Graves-Morris (cd.). Pude Approxi- Approximants, Inst. of Phys., Bristol, pp. 132-133. • Gammci. J. L. A974); "Continuation of functions beyond natural boundaries," Rocky Mtn. J. Math., 4, 203-206. Ciammcl,'J. L. and Nuttall, J. A973), "Convergence of Padc approximants to quasi-analytic functions beyond natural boundaries,",7. Math. Anal. App!., 43, 694-696. Ciilewicz, J. A978), Approximants de Padc, Springer Lecture Notes on Mathematics. No. 667, pp..309, 396. Pindor, M. A979), "Padc approximants and rational functions as tools for finding poles and zeros of analytical functions measured experimentally," in L. Wuytack (cd.). Pads Approximation and Its Applications, Springer Lecture Notes in Mathematics, No: 765, pp; 338-347. Избранная литература по различным приложениям аппроксимаций Паде и близким темам Течение жидкости Van Dyke. M. (,1958), "The supersonic blunt body problem—review and extension,"/, of Aerospace Sciences, 25, 485-496. VanTuyl. A. H. A971), "Use of Padc fractions in the calculation of blunt body flow's,"/'/'''* j., 9, 1431-1433. Van Tuyl. A. H. A973). "Calculation of nozzle flows using Padc fractions," A1AA J., II, 537-541. Кинетика Devooghl, J. and Muinl. E. (W5), "Now developments in three-dimensional neutron kinetics and review of kinetic benchmark calculations." in Рпч: of Specialists Meeting, (ian-hing Confirmee. Nuel. Energy Agency Committee on Reactor Physics. Пространства с индефинитным внутренним произведением Van Rossinn, H. A953), A Tlieoiy of Orthogonal Polynomials Based on the Padc Table, Van Ciorcum, Netherlands. Van Rossum. H. A979), "Orthogonal expansions in indefinite inner product spaces," in L. Wuytack (ed.). Padc Approximation and Its Applications, Springer Lecture Notes in Mail)cmatic,4, No. 765, Berlin, pp. 172-183.
Вычисление матричных экспонент Fair, W. and I.ufcc, Y. I.. A970), "Pade approximams to (Tic operator exponential," Num. Math.. 14.379-3K2. Molor, С and Van Loan, С A978), "Nineteen dubious ways to compute the exponential оГ а matrix," SI AM Review, 20, 801-836. Wragg, Л. and Davies, C. A975), "Computation of the exponential of a matrix II. Praetieal considerations,"./, hist. Math. Appl., 15, 273-278. Молекулярная поляризуемость Barnsley, M. F. A975), "Bounds on dynamical polarizabilities at high frequencies," Molecular Physics ,29, 1377-1386. Corcoran, С. Т. and Langhoff, P. W. A977), "Moment theory approximations for non-negative spectral densities," J. Math. Phys., 18, 651-657. Langhoff, P. W. and Karplus, M. A970), "Application of Pade approximants to dispersion force and optical polarizability computations," in G. A. Baker, Jr., and J. L. Gammel (cds.), The Pade Approximant in Theoretical Physics, Academic Press, New York, pp. «1-97. Перколяция Dunn, А. С Essam, J. W., and Ritchie, D, S. A975), "Scries expansion study of the pair connectedness in bond percolation models," J. Pliy's. C, 8, 4219- 4235. Функции Громмера Edrei, A. A975), "The complete Pade tables of certain series of simple fractions," Rocky Mm. J. Math., 5,559-584. Соли тоны Lambert. F. A980), "Pade approximanls and closed form solutions of the KdV and MKdV equations," Zeit fur Physik (Particles and Fields), 5, 147-150. Turchctti, G. A980), "Pade approximants and soliton solutions of the KdV equation," Lett NuovoCim.,27,1*07—110. Турбулентность Kraichnan, R. H. A970), "Turbulent diffusion: evaluation of primitive and renormalised perturbation series by Pade approximants and by expansion of Sticltjes transforms into contributions from continuous orthogonal functions" in G. A. Baker^Jr. and 1. L. Gammel (eds.). The Pa i approximant in theoretical physics. Academic Press, New York, pp 129-170.
Литература, имеющаяся на русском языке Ахиезер Н. И. Классическая проблема моментов.— М.: Физматгиз, 1961. Гренандер У., Cere Г. Теплицевы формы и их приложение.— М.: Изд-во Иностр. лит-ры, 1961. Джоунс У., Трон У. Непрерывные дроби. Аналитическая теория и приложе- приложения.—М.: Мир, 1985. Дирак П. Лекции по квантовой механике.— М.: Мир, 1968. Курант Р., Гильберт Д. Методы математической физики.— М.: Гостехиздат, 1951. Люк Ю. Специальные математические функции и их аппроксимации. Пер. с англ.—М.: Мир, 1980. Марков А. А. Избранные труды по теории непрерывных дробей и теории функ- функций, наименее уклоняющихся от нуля.— М.: Гостехиздат, 1948. Рид М., Саймон Б. Методы современной математической физики. Т. 3. Теория рассеяния.— М.: Мир, 1982. Рисе Ф.,Секефальви-Надь Б. Лекции по функциональному анализу.— М.: Мир, 1979. Рудин У. Основы математического анализа.— М.: Мир, 1966. Стильтьес Т. И. Исследования о непрерывных дробях.— Харьков—Киев: ОНТИ, 1936. Титчмарш Е. Теория функций.— М.: Наука, 1980. Трикоми Ф. Интегральные уравнения.—М.: Изд-во иностр. лит-ры, I960. Уолш Дж. Л. Интерполяция и аппроксимация рациональными функциями в комплексной области.—М.: Изд-во иностр. лит-ры, 1961. Харди Г. Расходящиеся ряды.— М.: Изд-во Иностр. лит-ры, 1951. Хованский А. Н. Приложение цепных дробей и их обобщении к вопросам приб- приближенного анализа.— М.: Гостехиздат, 1956. Чебышёв П. Л. Полное собрание сочинений, т. III.— М.— Л.: Издательство АН СССР, 1948. Эрдейи А. Асимптотические разложения,— Пер. о англ.— М.: Физматгиз, 1962. Литература на русском языке, добавленная при переводе Аптекарев А. И. A980) Асимптотика определителей Адамара и сходимость строк аппроксимаций Паде для суммы экспонент.— Матем. сборник, т. 113 A55), № 4, с. 520— 537. Буслаев В. И. A982) О полюсах m-й строки таблицы Паде.— Матем. сборник, т. 117 A59), с. 435—441. Буслаев В. И., Гончар А. А., Суетин С. П. A983) О сходимости подпоследовательностей /n-й строки таблицы Паде.— Ма- Матем. сборник, т. 120 A62), № 4, с. 540—545. Вавилов В. В. A976) Об особых точках мероморфной функции, заданной своим рядом Тейло- Тейлора.—ДАН СССР, т. 231, Л° 6, с. 1281—1284. A976) О сходимости аппроксимаций Паде мероморфных функций.— Матем. сборник, т. 101 A43), № I, с. 44—56. Вавилов В. В., Лопес Г., Прохоров В. А. A979) Об одной обратной задаче для строк таблицы Паде.— Матем. сборник, т. ПО A52), № I, с. 117—129. Вавилов В. В., Прохоров В. А., Суетин С. П.
494 Литератур* A983) Полюсы m-й строки таблицы Паде н особые точки функции.— Матем. сборник, т. 122 A64), № 4, с. 475—480. Гончар Л. А. A973) О сходимости аппроксимаций Паде.—Матем. сборник, т. 92 A34), № 1( стр. 152—164. A974) Локальное условие однозначности аналитических функций нескольких переменных.— Матем. сб., т. 93 A35), с. 296—313. A975) О сходимости аппроксимаций Паде для некоторых классов мероморфных функций.— Матем. сборник, т. 97 A39), № 4, стр. 607—629. A975) О сходимости обобщенных аппроксимаций Паде мероморфных функ- функций,—Матем. сборник, т. 98 A40), № 4, стр. 564—577. A981) Полюсы строк таблицы Паде и мероморфмое продолжение функций.— Матем. сборник, т. 115 A57), № 4, стр. 590—613. A982) О равномерной сходимости диагональных аппроксимаций Паде.— Ма- Матем. сборник, т. 118 A60), № 4, стр. 535—555. A984) О скорости рациональной аппроксимации аналитических функций.— Тру- Труды МИАН, т. 166, с. 52—60. Гончар А. А., Гиермо Лопес Л. A978) О теореме Маркова для многоточечных аппроксимаций Паде.— Матем. сборник, т. 105 A47), № 4, стр. 512 — 524. Гончар А. А., Рахманов Е. А. A981) О сходимости совместных аппроксимаций Паде для систем функций мар- марковского типа.— Труды МИАН, т. 157, с. 31—48. A984) Равновесная мера и распределение нулей экстремальных многочленов.— Матем. сб., т. 125 A67), с. 117—127. Дзядык В. К. A979) Об асимптотике диагональных аппроксимаций Паде функций sin г, cos г, shz и ch г. — Матем. сборник, т. 108 A50), № 2, с. 247—267. Дзядык В. К., Филозоф Л. И. A978) О скорости сходимости аппроксимаций Паде для некоторых элементар- элементарных функций.—Матем. сборник, т. 107 A49), № 3, с. 347—363. Калягин В. А. A980) Замечание о структуре таблицы Паде.— Вестник Московского ун-та, сер. 1. Математика, механика, № 5, с. 38—41. Ковачева Р. К- A979) О рациональных аппроксимациях мероморфных функций,— ДАН СССР, т. 211, № 3, с. 540—543. Лопес Г. A978) Условия сходимости многоточечных аппроксимаций Паде для функций стильтьесовского типа.— Матем. сб., т. 107 A49), с. 69—83. Никишин Е. М. A976) О сходимости диагональных аппроксимаций Пале для некоторых функ- функций.—Матем. сборник, т. 10! A43), № 2, с. 280—292. A979) О системе марковских функций.— Вестник Московского университета. Сер. 1, Математика и механика, № 4, с. 60—63. A980) О совместных аппроксимациях Паде.— Матем. сб., т. 113 A55), с. 499—519. Парусников В. И. A981) Алгоритм Якоби — Перрона и совместное приближение функций.— Матем. сб., т. 114 A56), с. 322—334. Рахманов Е. А. A977) О сходимости диагональных аппроксимаций Паде.— Матем. сборник, т. 104 A46), № 2, стр. 271—291. A980) О сходимости аппроксимаций Паде в классах голоморфных функций.— Матем. сборник, т. [12 A54), № 2, с. 162—169. Суетин С. П. A978) О сходимости рациональных аппроксимаций полиномиальных разложе-
Литература 495 ний в областях мероморфности заданной функции.— Матем. сб., т. 105 A47), с. 413—430. A979) Обратные теоремы об обобщенных аппроксимациях Паде.— Матем. сб., т. 109 A51), с. 629—645. A981) О теореме Монтессу де Болора для рациональных аппроксимаций орто- ортогональных разложений.— Матем. сб., т. 114 A56), с. 451—464. A983) О полюсах m-й строки таблицы Паде.—Матем. сборник, т. 120 A62), № 4, с. 500-504. A984) Об одной обратной задаче для т-й строки таблицы Паде.—Матем. сбор- сборник, т. 124 A66), N° 2, с. 238—250.
ПРЕДМЕТНЫЙ УКАЗАТЕЛЬ Алгоритм Бейкера 76 — биортогональный 358 — Висковатова 119, 134—140 — Евклида 76, 138 — Кленшоу — Лорда 341—343 — Кронекера 293—295 — надежность 73, 136, 293 — Тзчера — Тьюки 296—298 — Q.D. 108, 128—130 для Т-дробей 305 обобщенный 298 — е 85—89, 39—98 для векторных последователь- последовательностей 338 обобщенный 299—301 — т) 89—93 Алгоритмы для рациональной интерпо- интерполяции 293—301 Анализ рядов 315—323 Ангармонический осциллятор 448— 449 Аппроксимации Бейкера — Гаммеля 305—315 — для случая многих переменных 323—333 — Паде— Бореля 313 — Паде вариационные 392—399 знаменатель 16 матричные 333—339 многоточечные 248, 287—305 определение 30, 31 — Паде — Фурье 344 — Паде — Чебышёва 339—344 числитель 17 — Фишера 330 — Чисхолма 324—328 — Эрмита— Паде 319 Асимптотическая сходимость 197— 198 Ассоциированная непрерывная дробь 131 Биградиенты 47—51, 139 — полиномиальные 47—48 Биномиальная функция 142, 146 Блоки 33—41 Вариационный принцип (метод) Бесси- са 381-382 Кона 377—379 Рзлея — Ритца 373—374, 380 Швингера 379 Вещественно-симметричная функция 160-161 Вронскиан 368 Гамма-функция 200—202 Гауссова квадратура 405—410 Гиперболический тангенс 142, 146 Гипергеометрическая функция 144— 148 - Л(.) 52, 144 -Л(-) 52, 145 - 2Fn(.) 52, 144, 195 Гипотеза Бейкера, Гаммеля и Уиллса 276 Двойственность 42 Детерминаитные неравенства для коэф- коэффициентов ряда Гамбургера 205—206 Стильтьеса 162—163, 176 — тождества для коэффициентов ряда Стильтьеса 165—166 — формулы для аппроксимаций Паде 16-19, 53—56 Дефект 63 Диагональные последовательности 230 Диффузионные процессы 423—431 Дополнительная функция ошибок 230 Г-дроби 130
Предметный указатель 497 — общего вида 133 — регулярные 130 /-дроби 132, 221 Р-дроби 130 S-дроби 131, 165, 203—204 — для рядов Стилыпьеса 165, 103— 204 Г-дроби 141, 302—305 Дробно линейная инвариантность 42— 43 Емкость 267—276 Звезда Миттаг-Леффлера 58 «Звездное» тождество 33 Интеграл Доусона 144 — обобщенный 304 Интегральная показательная функция первого порядка 184 — — — разложение в непрерывную дробь 143 Интегральные уравнения 346—360 Интерполяционные полиномы Ньютона 288—289 Квазианалитические функции 62—64 Коллокация 416—422 Компактная форма Н'атолла для аппро- аппроксимаций Паде 26 Контрпример Гаммеля 277 — Перрона 233 Критерий Карлемана для рядов Гам- Гамбургера 220 — Стилыпьеса 198—199 приложения 447 Лемма Картина 267, 275 — Шварца 187—188 Лемнискаты 267—271 Логарифмическая емкость, трансфи- трансфинитный диаметр 275 Лучевые последовательности 189 Матрица Ганкеля, число обусловлен- обусловленности 75 /С-матрица 366, 387 S-матрица 387 Г-матрица 363, 387 Мера, сходимость по мере 257 а-мера Хаусдорфа 275 Модель Пири 445 Моменты Гамбургера 205 — оценки для них 410—416 Стилыпьеса 159, 177 Хаусдорфа 193 Метод Висковатова 119, 134—140 — Йоста 399 — Кранка — Нжольсона 423 — критической точки 456—316 — Лагерра 441—443 — Ланцоша 416—358 — обратных разностей Тиле 295 рекуррентных соотношений 120 — отношений 316 — прямых рекуррентных соотношений 119—120 т-метод Ланцоша 416 А2-метод Эйткена 79—83 Методы гильбертовых пространств 355—361 Натуральный логарифм 143, 146 Неполная гамма-функция 144 Непрерывные дроби ПО—157 ассоциирование 131 определение 1I1 периодические непрерывные дроби 122 подходящие дроби 111 — — рекуррентные соотношения 112—113 — — сжатие непрерывной дроби 131 условие расходимости 150 — — формула суммирования 121 — — эквивалентное преобразование 112 — — элементы непрерывной дроби 111 Неравенства Чебышёва для функции плотности 410—416 Определение Фробениуса 30 Определенность проблемы моментов 177-178, 192 Определитель Вандермонда 239—241 — Ганкеля 18, 50 — C(UM) 32 — O(m, я) 162 Ортогональные полиномы 181—184, 206, 249 Основная неполная гамма-функция 143 Парадиагональные последовательности 41, 185-187, 231 Парциальная волна Пион-пион рассеяние 450—454
498 Предметный указатель Полиномы Лагерра 184 — Чебышёва 267—271 Порядок аппроксимации 12—14, 287 Последовательность аппроксимаций Паде взятая по столбцу 41 — — — диагональная 27, 42 — — — лучевая 189 парадиагональная 41, 185— 187, 231 строчная 41 — Штурма 167 Построение аппроксимаций Паде алгебраические методы 24—29, 52— 57, 76—78 — — — численные методы 71—78 Потенциалы знакопостоянные 385— 392 — сингулярные 399—405 Потенциальное рассеяние 361—369 Правило ромба 127—129 Преобразование Лапласа 431—433 Проблема Коши — Якоби 287 — Маркова 415 — корней 103—109 — моментов 177—178 хаусдорфова 192 — — целочисленная 193 Равностепенная непрерывность 173 Разделенные разности 288 Разложение по сильным полям 458 Разложение Эйлера в непрерывную дробь 140 Рациональная аппроксимация 305—315 — интерполяция 287—305 алгоритмы построения 293—315 Ряд Гамбургера 205 — Грегори 37 — Стильтьеса 158 Свойство чередования нулей 168—169, 412 Таблица Паде 19, 37—42 С-таблица 29—42 Q.D.-таблица 128—129 Тангенс 142 Тёплицевы матрицы 77—78 Теорема Адамара об определителях 50 — АрЦела 174 — Бейкера, Гаммеля и Уиллса 43 — Бирдона 232 — Ван Флека 153 — Веперштрасса о равномерной схо- сходимости 175 о существенно особой точке 60 — Гамбургера 217 — Лорана 57 — Монтессу 234—248 обобщенная 248, 439 — Натолла 262 — о параболе для непрерывных дробей \о2. — Поммеренке 204 — Рунге 435 — Саффа и Варги о параболе 225 — Сильвестра 33—34 — Труди 51 — Уолша 439 — усечения 45 — Эйлера о рекуррентных соотноше- соотношениях 112 Теория рассеяния 361—370 Тождество Винна 93 для соседних аппроксимаций 98—103 — Корделье 96 Унитарность 45—46, 326, 336 Уравнение Бете — Салпитера 386, 391 — Липпмана — Швингера 363, 391 — Риккати 440 Уравнения Паде 13—14 Ускорение сходимости 24—29 Л-уетойчивость 529 /.-устойчивость 529 Формула Коши — Бине 235 — обращения 217 — остатка для аппроксимаций Паде 245 ряда Стильтьеса 183—184,406— 407 — суммирования 121 Эйлера — Маклорена 408 — Эрмита 244—246 Функция Гамбургера 205 — Герглотца 222 — Грина 369 для решения Поста 370 стоячих волн 369, 388 S-волны 379 свободная 379 — и ряд Эйлера 28, 198 — Ле Руа 315 — ошибок 143, 304 — плотности 178—180 — Стильтьеса 158
Предметный указатель 499 Характеристическая функция 406—407 разложения в непрерывные Хаусдорфова проблема моментов 192 дроби 142 теоремы Саффа и Варги 225— 227 Частотные ряды Попа 223—230 Ядра вполне непрерывные 351 Экспоненциальная функция аппрок- — компактные 35! симации Наде 19—24 — конечномерные 347
ОГЛАВЛЕНИЕ ОТ ИЗДАТЕЛЬСТВА 5 ОТ РЕДАКТОРА ЭНЦИКЛОПЕДИИ 7 ПРЕДИСЛОВИЕ РЕДАКТОРА СЕРИИ 8 ПРЕДИСЛОВИЕ 10 ЧАСТЬ 1. ОСНОВЫ ТЕОРИИ 12 Глава 1. ВВЕДЕНИЕ И ОПРЕДЕЛЕНИЯ 12 1.1. Введение и основные понятия 12 1.2. Аппроксимации Паде экспоненциальной функции 19 1.3. Последовательности и ряды. Трудности 24 1.4. Определение Бейкера, с-таблица и блочная структура ... 29 1.5. Двойственность и инвариантность 42 1.6. Биградиенты и формула Адамара 47 Глава 2. ПРЯМЫЕ ПРИЛОЖЕНИЯ 52 2.1. Прямой метод нахождения аппроксимаций Паде 52 2.2. Распознавание особенпостеi с помощью аппроксимаций Паде 57 2.3. Предполагаемые ошибки 67 2.4. Численные методы нахождения аппроксимаций Паде .... 71 Глава 3. АППРОКСИМАЦИИ ПАДЕ И ЧИСЛЕННЫЕ МЕТОДЫ ... 79 3.1. Д2-метод Эйткена как метод аппроксимаций Паде [L/\] ... 79 3.2. Ускорение и сверхускорение сходимости 83 3.3. е-алгоритм и т]-алгоритм 85 3.4. Тождество Випна и е-алгоритм 93 3.5. Общие тождества и рекуррентные формулы 98 3.6. Q. D.-алгоритм и проблема корней 103 Глава 4. СВЯЗЬ С НЕПРЕРЫВНЫМИ ДРОБЯМИ НО 4.1. Определения и рекуррентные соотношения ПО 4.2. Непрерывные дроби, связанные с рядом Тейлора 114 4.3. Алгебраические и численные методы , 118 4.4. Различные представления непрерывных дробей 122 4.5. Типы непрерывных дробей 130 4.5.1. Регулярные дроби и невырожденных случаях .... 130 4 5.2. Дроби общего вида в вырожденных случаях 132 4.5.3. Алгоритм Висковатова в общем случае 134 4.5.4. Некоторые специальные типы непрерывных дробей . , 140
Оглавление 501 4.6. Примеры непрерывных дробей, являющихся аппроксимациями Паде 141 4.7. Сходимость непрерывных дробей 149 Глава 5. РЯДЫ СТИЛЬТЬЕСА И РЯДЫ ПОЙА 158 5.1. Ряды Стильтьеса; введение 158 5.2. Сходимость рядов Стильтьеса 166 5.3. Проблема моментов и ортогональные полиномы 177 5.4. Ряды Стильтьеса, сходящиеся в круге \г\< R 185 5.4.1. Хаусдорфова проблема моментов 192 5.4.2. Целочисленная проблема моментов 193 5.5. Расходящиеся ряды Стильтьеса 195 5.6. Ряды Гамбургера и проблема моментов Гамбургера ..... 205 5.7. Частотные ряды Пойа 223 Глзва 6. ТЕОРИЯ СХОДИМОСТИ 231 6.1. Введение 231 6.2. Теорема Монтессу 234 6.3. Формула Эрмита и теорема Монтессу 244 6.4. Единственность предела 250 6.5. Сходимость по мере 257 6.6. Лемнискаты, емкость и мера 267 6.7. Паде-гипотеза 276 Приложение. ПРОГРАММА НА ФОРТРАНЕ 281 Часть 2. ОБОБЩЕНИЯ И ПРИЛОЖЕНИЯ 287 Глава 1. ОБОБЩЕНИЯ АППРОКСИМАЦИЙ ПАДЕ 287 1.1. Многоточечные аппроксимации Паде 287 1.2. Аппроксимации Бейксра — Гаммеля 305 1.3. Анализ рядов 315 1.4. Аппроксимация функций нескольких переменных 323 1.5. Матричные аппроксимации Паде 333 1.6. Аппроксимации Паде — Чебышёва, Паде — Фурье и т. д. . . 339 Глава 2. СВЯЗЬ С ИНТЕГРАЛЬНЫМИ УРАВНЕНИЯМИ И КВАНТО- КВАНТОВОЙ МЕХАНИКОЙ 346 2.1. Общий метод и конечномерные ядра 346 2.2. Аппроксимации Паде и интегральные уравнения с компактны- компактными ядрами 349 2.3. Методы, основанные на проектировании 355 2.4. Потенциальное рассеяние 361 2.5. Связь аппроксимаций Паде с вариационными принципами . . 372 2.6. Оценка погрешности аппроксимаций Паде в вариационных принципах 383 2.7. Знакопостоянные потенциалы в теории рассеяния и т. д. . . 385 2.8. Вариационные аппроксимации Паде 392 2.9. Сингулярные потенциалы 399 Глава 3. СВЯЗЬ С ЧИСЛЕННЫМ АНАЛИЗОМ 405 3.1. Квадратура Гаусса 405 3.2. Неравенства Чебышёва для функции плотности 410 3.3. Метод коллокации и т-метод 416 3.4. Метод Кранка — Никольсона и близкие к нему методы реше- решения уравнения диффузии . , _ 423
502 Оглавление 3.5. Обратное преобразование Лапласа 431 3.6. Связь с рациональной аппроксимацией 433 3.7. Аппроксимации Паде для уравнения Риккати 440 Глава 4. СВЯЗЬ С КВАНТОВОЙ ТЕОРИЕЙ ПОЛЯ 444 4.1. Возмущенные гармонические осцилляторы 444 4.2. Пион-пион рассеяние 450 4.3. Решеточное обрезание в Яф4-евклидовой теории поля или мо- модель Изинга непрерывного спина 455 ЛИТЕРАТУРА 460 ПРЕДМЕТНЫЙ УКАЗАТЕЛЬ 496
Монография Джордж Бейкер мл., Питер Грейвс-Моррис АППРОКСИМАЦИИ ПАДЕ Ст. научный редактор А. А. Бряндинская. Млад, научн. ред. Н. С. Полякова, Т. А. Денисов; Художник Н. Ф. Алексеев. Художественный редактор В. И. Шаповалов Технический редактор И. М. Кренделева. Корректор Н. А. Мистрюкова ИБ № 5415 Сдано в набор 26.06.85. Подписано к печати 11.06.86. Формат 60x90l/ie- Бумага книжн.-журн. Печать высокая. Гарнитура латинская. Объем 15,75 бум. л. Усл. печ. л. 31,50. Усл. кр.-отт. 31,50. Уч.-изд. л. 29,62. Изд. № 1/3846. Тираж 6 400 экз. Заказ Л"» ;228Г. Цена 4 р. 80 к. ИЗДАТЕЛЬСТВО «МИР». 129820, ГСП, Москва, И-110, 1-й Рижский пер., 2 Ордена Октябрьской Революции и ордена Трудового Красного Знамени МПО «Первая Образцовая типография» имени А. А. Жданова Союзполиграфпрома при Государственном комитете СССР по делам издательств, полиграфии и книжной торговли. 113054. Москва, Валовая, 28 Отпечатано с матриц а Ленинградской типографии № i ордена Трудовою Красного Знамени Ленинградского ооъединения «Техническая книга» им. Евгении Соколовой Союзполш рафпрома при Государственном коми- комитете СССР по делам издательств,^ полиграфии и книжной торговли. 191126, Ленинград,"Социалистическая ул., 14.


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