Padé–Chebyshev approximants of multivalued analytic functions, variation of equilibrium energy, and the $S$-property of stationary compact sets

A. A. Gonchar   ; E. A. Rakhmanov   ; S. P. Suetin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 66 (2011) no. 6, pp. 1015-1048

Résumé

Padé–Chebyshev approximants are considered for multivalued analytic functions that are real-valued on the unit interval $[-1,1]$. The focus is mainly on non-linear Padé–Chebyshev approximants. For such rational approximations an analogue is found of Stahl's theorem on convergence in capacity of the Padé approximants in the maximal domain of holomorphy of the given function. The rate of convergence is characterized in terms of the stationary compact set for the mixed equilibrium problem of Green-logarithmic potentials. Bibliography: 79 titles.

Détail

Keywords: rational approximation, Chebyshev polynomials, non-linear Padé–Chebyshev approximants, stationary compact set, Stahl's theorem, convergence in capacity.
Mots-clés : Padé approximants

Bibliographie

[1] J. L. Walsh, Interpolation and approximation by rational functions in the complex domain, Amer. Math. Soc. Colloq. Publ., XX, 3rd ed., Amer. Math. Soc., Providence, RI, 1960, x+398 pp. | MR | MR | Zbl | Zbl

[2] A. A. Gončar, “On the speed of rational approximation of some analytic functions”, Math. USSR-Sb., 34:2 (1978), 131–145 | DOI | MR | Zbl | Zbl

[3] H. Stahl, “Extremal domains associated with an analytic function. I”, Complex Variables Theory Appl., 4:4 (1985), 311–324 | DOI | MR | Zbl

[4] H. Stahl, “Extremal domains associated with an analytic function. II”, Complex Variables Theory Appl., 4:4 (1985), 325–338 | DOI | MR | Zbl

[5] H. Stahl, “Structure of extremal domains associated with an analytic function”, Complex Variables Theory Appl., 4:4 (1985), 339–354 | DOI | MR | Zbl

[6] H. Stahl, “Orthogonal polynomials with complex valued weight function. I”, Constr. Approx., 2:3 (1986), 225–240 | DOI | MR | Zbl

[7] H. Stahl, “Orthogonal polynomials with complex valued weight function. II”, Constr. Approx., 2:3 (1986), 241–251 | DOI | MR | Zbl

[8] H. Stahl, “Convergence of rational interpolants”, Bull. Belg. Math. Soc. Simon Stevin, 1996, suppl. issue – Numerical analysis. Papers from the Conference on Special Topics in Numerical Analysis and Applied Mathematics in honor of Jean Meinguet (Louvain-la-Neuve, 1995), 11–32 | MR | Zbl

[9] H. Stahl, “The convergence of Padé approximants to functions with branch points”, J. Approx. Theory, 91:2 (1997), 139–204 | DOI | MR | Zbl

[10] A. A. Gonchar, E. A. Rakhmanov, “Equilibrium distributions and degree of rational approximation of analytic functions”, Math. USSR-Sb., 62:2 (1989), 305–348 | DOI | MR | Zbl | Zbl

[11] A. I. Aptekarev, V. I. Buslaev, A. Martines-Finkelshtein, S. P. Suetin, “Approksimatsii Pade, nepreryvnye drobi i ortogonalnye mnogochleny”, UMN, 66:6 (2011), 37–122

[12] R. Jentzsch, “Untersuchungen zur Theorie der Folgen analytischer Funktionen”, Acta Math., 41:1 (1916), 219–251 | DOI | MR | Zbl

[13] G. Szegő, “Über die Nullstellen von Polynomen, die in einem Kreise gleichmässig konvergieren”, Sitzungsber. Berl. Math. Ges., 21 (1922), 59–64 | Zbl

[14] D. V. Khristoforov, “On asymptotic properties of interpolation polynomials”, Math. Notes, 83:1 (2008), 116–124 | DOI | MR | Zbl

[15] E. A. Lebedeva, “A generalization of Jentzsch's theorem”, Math. Notes, 88:5 (2010), 717–722 | DOI | Zbl

[16] H.-P. Blatt, R. K. Kovacheva, “Growth behavior and zero distribution of rational approximants”, Constr. Approx., 34:3 (2011), 393–420 | DOI

[17] E. Remes, “Sur le calcul effectif des polynômes d'approximation de Tschebyscheff”, C. R. Acad. Sci. Paris, 199 (1934), 337–340 | Zbl

[18] N. I. Achieser, Theory of approximation, Dover Publications, Inc., New York, 1992, x+307 pp. | MR | MR

[19] N. I. Akhiezer, “Chebyshevskoe napravlenie v teorii approksimatsii”, Matematika XIX veka, Vyp. 3, eds. A. N. Kolmogorov, A. P. Yushkevich, Nauka, M., 1987, 9–79 | MR | Zbl

[20] E. Ya. Remez, Osnovy chislennykh metodov chebyshevskogo priblizheniya, Naukova dumka, Kiev, 1969, 624 pp. | MR | Zbl

[21] V. I. Lebedev, “Finding polynomials of best approximation with weight”, Sb. Math., 199:2 (2008), 207–228 | DOI | MR | Zbl

[22] V. I. Lebedev, “O trigonometricheskoi forme chebyshevskikh teorem ob alternanse i fazovom iteratsionnom metode nakhozhdeniya nailuchshikh s vesom priblizhenii”, Ufimsk. matem. zhurn., 1:4 (2009), 110–118 | Zbl

[23] A. B. Bogatyrev, “Effective solution of the problem of the optimal stability polynomial”, Sb. Math., 196:7 (2005), 959–981 | DOI | MR | Zbl

[24] A. B. Bogatyrev, “Chebyshev representation for rational functions”, Sb. Math., 201:11 (2010), 1579–1598 | DOI | MR

[25] J. M. Borwein, M. P. Skerritt, An introduction to modern mathematical computing. With Maple$^{\rm TM}$, Springer Undergrad. Texts Math. Technol., Springer, New York, 2011, xvi+216 pp. | MR | Zbl

[26] O. B. Arushanyan, N. I. Volchenskova, S. F. Zaletkin, “Priblizhennoe reshenie obyknovennykh differentsialnykh uravnenii s ispolzovaniem ryadov Chebysheva”, Sib. elektron. matem. izv., 7 (2010), 122–131 | MR

[27] O. B. Arushanyan, N. I. Volchenskova, S. F. Zaletkin, “O vychislenii koeffitsientov ryadov Chebysheva dlya reshenii obyknovennykh differentsialnykh uravnenii”, Sib. elektron. matem. izv., 8 (2011), 273–283

[28] J. P. Boyd, Chebyshev and Fourier spectral methods, 2nd ed., Dover Publications, Mineola, NY, 2001, xvi+668 pp. | MR | Zbl

[29] J. P. Boyd, “Chebyshev expansion on intervals with branch points with application to the root of Kepler's equation: a Chebyshev–Hermite–Padé method”, J. Comput. Appl. Math., 223:2 (2009), 693–702 | DOI | MR | Zbl

[30] S. P. Suetin, “On de Montessus de Ballore's theorem for nonlinear Pade approximants of orthogonal expansions and Faber series”, Soviet Math. Dokl., 22 (1980), 274–277 | MR | Zbl

[31] K. O. Geddes, “Block structure in the Chebyshev–Padé table”, SIAM J. Numer. Anal., 18:5 (1981), 844–861 | DOI | MR | Zbl

[32] S. P. Suetin, “On the existence of nonlinear Padé–Chebyshev approximations for analytic functions”, Math. Notes, 86:2 (2009), 264–275 | DOI | MR | Zbl

[33] L. A. Knizhnerman, “Padé–Faber approximation of Markov functions on real-symmetric compact sets”, Math. Notes, 86:1 (2009), 81–92 | DOI | MR | Zbl

[34] G. A. Baker, Jr., P. Graves-Morris, Padé approximants. Part I. Basic theory, Encyclopedia of Mathematics and its Applications, 13, Addison-Wesley, Reading, MA, 1981, xx+325 pp. ; Padé approximants. Part II. Extensions and applications, Encyclopedia of Mathematics and its Applications, 14, Addison-Wesley, Reading, MA, 1981, xviii+215 pp. | MR | MR | Zbl | Zbl | Zbl

[35] A. A. Gonchar, E. A. Rakhmanov, S. P. Suetin, “On the convergence of Padé approximation of orthogonal expansions”, Proc. Steklov Inst. Math., 200 (1993), 149–159 | MR | Zbl

[36] A. A. Gonchar, E. A. Rakhmanov, S. P. Suetin, “On the rate of convergence of Padé approximants of orthogonal expansions”, Progress in approximation theory (Tampa, FL, 1990), Springer Ser. Comput. Math., 19, Springer, New York, 1992, 169–190 | MR | Zbl

[37] A. A. Gonchar, E. A. Rakhmanov, S. P. Suetin, O skhodimosti approksimatsii Chebysheva–Pade dlya veschestvennykh algebraicheskikh funktsii, 2010, 10 pp., arXiv: 1009.4813

[38] A. A. Gonchar, E. A. Rakhmanov, S. P. Suetin, Skhodimost nelineinykh approksimatsii Pade–Chebysheva dlya mnogoznachnykh analiticheskikh funktsii, variatsiya ravnovesnoi energii i $S$-svoistvo statsionarnykh kompaktov, 2010, 49 pp., arXiv: 1012.0170

[39] A. A. Gonchar, E. A. Rakhmanov, S. P. Suetin, “O skhodimosti approksimatsii Pade–Chebysheva dlya analiticheskikh funktsii s konechnym chislom tochek vetvleniya”, Dokl. RAN, 2012 (to appear)

[40] O. L. Ibryaeva, “Dostatochnoe uslovie edinstvennosti lineinoi approksimatsii Pade–Chebysheva”, Izvestiya Chelyabinskogo nauchnogo tsentra, 2002, no. 4(17), 1–5 ; http://csc.ac.ru/ej/issue/ru/18 | MR

[41] R. S. Varga, Functional analysis and approximation theory in numerical analysis, CBMS-NSF Regional Conf. Ser. Appl. Math., 3, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1971, v+76 pp. | MR | MR | Zbl | Zbl

[42] R. S. Varga, “Scientific computation on some mathematical conjectures”, Approximation theory, V, Proceedings of the Fifth International Symposium on Approximation Theory (College Station, TX, 1986), eds. C. K. Chui, L. L. Schumaker, J. D. Ward, Academic Press, Boston, MA, 1986, 191–209 | MR | Zbl

[43] L. A. Knizhnerman, “Vydelenie polyusov potentsialnykh polei s pomoschyu razlozheniya v ryady Fure–Chebysheva”, Izv. AN SSSR. Ser. fizika Zemli, 1984, no. 11, 119–123

[44] G. L. Litvinov, “Error autocorrection in rational approximation and interval estimates. A survey of results”, Cent. Eur. J. Math., 1:1 (2003), 36–60 | DOI | MR | Zbl

[45] L. N. Trefethen, M. H. Gutknecht, “The Carathéodory–Fejér method for real rational approximation”, SIAM J. Numer. Anal., 20:2 (1983), 420–436 | DOI | MR | Zbl

[46] L. N. Trefethen, M. H. Gutknecht, “Padé, stable Padé, and Chebyshev–Padé approximation”, Algorithms for Approximation (Shrivenham, 1985), Inst. Math. Appl. Conf. Ser. New Ser., 10, Oxford University Press, New York, 1987, 227–264 | MR | Zbl

[47] Yu. A. Labych, A. P. Starovoitov, “Trigonometric Padé approximants for functions with regularly decreasing Fourier coefficients”, Sb. Math., 200:7 (2009), 1051–1074 | DOI | MR | Zbl

[48] L. M. Skvortsov, “Explicit stabilized Runge–Kutta methods”, Comput. Math. Math. Phys., 51:7 (2011), 1153–1166 | DOI

[49] S. Paszkowski, Zastosowania numeryczne wielomianów i szeregów Czebyszewa, Podstawowe Algorytmy Numeryczne, Pánstwowe Wydawnictwo Naukowe, Warsaw, 1975, 481 pp. (Polish) | MR | Zbl | Zbl

[50] V. I. Buslaev, “Simple counterexample to the Baker–Gammel–Wills conjecture”, East J. Approx., 7:4 (2001), 515–517 | MR | Zbl

[51] V. I. Buslaev, “On the Baker–Gammel–Wills conjecture in the theory of Padé approximants”, Sb. Math., 193:6 (2002), 811–823 | DOI | MR | Zbl

[52] H. Stahl, “Diagonal Padé approximants to hyperelliptic functions”, Ann. Fac. Sci. Toulouse Math. (6), 1996, Special issue, 121–193 | DOI | MR | Zbl

[53] S. P. Suetin, “Uniform convergence of Padé diagonal approximants for hyperelliptic functions”, Sb. Math., 191:9 (2000), 1339–1373 | DOI | MR | Zbl

[54] D. V. Khristoforov, “On uniform approximation of elliptic functions by Padé approximants”, Sb. Math., 200:6 (2009), 923–941 | DOI | MR | Zbl

[55] N. S. Landkof, Foundations of modern potential theory, Grundlehren Math. Wiss., 180, Springer-Verlag, Berlin–Heidelberg–New York, 1972, 424 pp. | MR | MR | Zbl | Zbl

[56] A. A. Gonchar, E. A. Rakhmanov, “Equilibrium measure and the distribution of zeros of extremal polynomials”, Math. USSR-Sb., 53:1 (1986), 119–130 | DOI | MR | Zbl

[57] A. Martines-Finkelshtein, E. A. Rakhmanov, S. P. Suetin, “Variatsiya ravnovesnoi energii i $S$-svoistvo statsionarnykh kompaktov”, Matem. sb., 202:12 (2011), 113–136

[58] K. M. Ermokhin, “Tekhnologiya postroeniya razrezov metodom analiticheskogo prodolzheniya geofizicheskikh polei”, Geoinformatika, 2010, no. 2, 51–60; http://www.geosys.ru/index.php/journal/archive.html

[59] A. A. Gonchar, E. A. Rakhmanov, “On the equilibrium problem for vector potentials”, Russian Math. Surveys, 40:4 (1985), 183–184 | DOI | MR | Zbl

[60] A. I. Aptekarev, “Asymptotics of Hermite–Padé approximants for two functions with branch points”, Dokl. Math., 78:2 (2008), 717–719 | DOI | MR | Zbl

[61] J. A. Jenkins, Univalent functions and conformal mapping, Ergeb. Math. Grenzgeb. Neue Folge, 18, Reihe: Moderne Funktionentheorie, Springer-Verlag, Berlin–Göttingen–Heidelberg, 1958, vi+169 pp. | MR | Zbl

[62] G. M. Goluzin, Geometric theory of functions of a complex variable, Translations of Mathematical Monographs, 26, American Mathematical Society, Providence, R. I., 1969, vi+676 pp. | MR | MR | Zbl | Zbl

[63] G. V. Kuz'mina, “Moduli of families of curves and quadratic differentials”, Proc. Steklov Inst. Math., 139 (1982), 1–231 | MR | MR | Zbl | Zbl

[64] A. Martínez-Finkelshtein, E. A. Rakhmanov, S. P. Suetin, “Variation of the equilibrium measure and the $S$-property of a stationary compact set”, Russian Math. Surveys, 66:1 (2011), 176–178 | DOI | Zbl

[65] G. V. Kuz'mina, “Gennadii Mikhailovich Goluzin and geometric function theory”, St. Petersburg Math. J., 18:3 (2007), 347–372 | DOI | MR | Zbl

[66] E. A. Perevoznikova, E. A. Rakhmanov, Variatsiya ravnovesnoi energii i $S$-svoistvo kompaktov minimalnoi emkosti, Preprint, M., 1994

[67] V. I. Buslaev, A. Martines-Finkelshtein, S. P. Suetin, Metod vnutrennikh variatsii i suschestvovanie $S$-kompaktov, 2012 (to appear)

[68] A. I. Aptekarev, “Sharp constants for rational approximations of analytic functions”, Sb. Math., 193:1 (2002), 1–72 | DOI | MR | Zbl

[69] A. Martínez-Finkelshtein, E. A. Rakhmanov, “On asymptotic behavior of Heine–Stieltjes and Van Vleck polynomials”, Recent trends in orthogonal polynomials and approximation theory, Contemp. Math., 507, Amer. Math. Soc., Providence, RI, 2010, 209–232 | MR | Zbl

[70] A. Martínez-Finkelshtein, E. Rakhmanov, “Critical measures, quadratic differentials, and weak limits of zeros of Stieltjes polynomials”, Comm. Math. Phys., 302:1 (2011), 53–111 | DOI | MR | Zbl

[71] A. A. Gonchar, E. A. Rakhmanov, “On the convergence of simultaneous Padé approximants for systems of functions of Markov type”, Proc. Steklov Inst. Math., 157 (1983), 31–50 | MR | Zbl | Zbl

[72] E. A. Rakhmanov, “The asymptotics of Hermite–Padé polynomials for two Markov-type functions”, Sb. Math., 202:1 (2011), 127–134 | DOI | MR | Zbl

[73] S. P. Suetin, “Nekotoryi analog variatsionnykh formul Adamara i Shiffera”, TMF, 2012 (to appear)

[74] A. I. Aptekarev, V. G. Lysov, D. N. Tulyakov, “Random matrices with external source and the asymptotic behaviour of multiple orthogonal polynomials”, Sb. Math., 202:2 (2011), 155–206 | DOI | MR

[75] A. I. Aptekarev, A. B. J. Kuijlaars, W. Van Assche, “Asymptotics of Hermite–Padé rational approximants for two analytic functions with separated pairs of branch points (case of genus $0$)”, Int. Math. Res. Pap. IMRP, 2008, Art. ID rpm007, 128 pp. | DOI | MR | Zbl

[76] V. I. Smirnov, N. A. Lebedev, Functions of a complex variable: Constructive theory, The M.I.T. Press, Cambridge, MA, 1968, 488 pp. | MR | MR | Zbl | Zbl

[77] S. W. Ellacott, “On the Faber transform and efficient numerical rational approximation”, SIAM J. Numer. Anal., 20:5 (1983), 989–1000 | DOI | MR | Zbl

[78] S. W. Ellacott, E. B. Saff, “Computing with the Faber transform”, Rational approximation and interpolation (Tampa, 1983), Lecture Notes in Math., 1105, Springer, Berlin, 1984, 412–418 | DOI | MR | Zbl

[79] A. A. Gonchar, G. López Lagomasino, “On Markov's theorem for multipoint Padé approximants”, Math. USSR-Sb., 34:4 (1978), 449–459 | DOI | MR | Zbl