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
жүктеу/скачать 343 Kb.
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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 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. 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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 ( 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. 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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 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. 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