Hermite-Padé approximation of exponential functions

A. V. Astafieva   ; A. P. Starovoitov

Sbornik. Mathematics, Tome 207 (2016) no. 6, pp. 769-791

Résumé

The paper is concerned with diagonal Hermite-Padé polynomials of the first kind for the system of exponentials $\{e^{\lambda_jz}\}_{j=0}^k$ with arbitrary distinct complex parameters $\{\lambda_k\}_{j=0}^k$. An asymptotic formula for the remainder term is established and the location of the zeros is described. For real parameters the asymptotics are found and the extremal properties are described. The theorems obtained supplement the well-known results due to Borwein, Wielonsky, Saff, Varga and Stahl. Bibliography: 43 titles.

Détail

Keywords: system of exponentials, asymptotic equalities, the Laplace method, the saddle-point method.
Mots-clés : Padé polynomials, Hermite-Padé polynomials

Bibliographie

[1] 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

[2] B. Beckermann, V. Kalyagin, Ana C. Matos, F. Wielonsky, “How well does the Hermite–Padé approximation smooth the Gibbs phenomenon?”, Math. Comp., 80:274 (2011), 931–958 | DOI | MR | Zbl

[3] V. N. Sorokin, “Cyclic graphs and Apéry's theorem”, Russian Math. Surveys, 57:3 (2002), 535–571 | DOI | DOI | MR | Zbl

[4] S. P. Suetin, “Padé approximants and efficient analytic continuation of a power series”, Russian Math. Surveys, 57:1 (2002), 43–141 | DOI | DOI | MR | Zbl

[5] S. P. Suetin, “Distribution of the zeros of Padé polynomials and analytic continuation”, Russian Math. Surveys, 70:5 (2015), 901–951 | DOI | DOI | MR

[6] P. M. Bleher, A. B. J. Kuijlaars, “Random matrices with external source and multiple orthogonal polynomials”, Int. Math. Res. Not., 2004:3 (2004), 109–129 | DOI | MR | Zbl

[7] A. I. Aptekarev, P. M. Bleher, A. B. J. Kuijlaars, “Large $n$ limit of Gaussian random matrices with external source. II”, Comm. Math. Phys., 259:2 (2005), 367–389 | DOI | MR | Zbl

[8] 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 | DOI | MR | Zbl

[9] V. A. Kalyagin, “Hermite–Padé approximants and spectral analysis of nonsymmetric operators”, Russian Acad. Sci. Sb. Math., 82:1 (1995), 199–216 | DOI | MR | Zbl

[10] A. I. Aptekarev, V. A. Kalyagin, E. B. Saff, “Higher-order three-term recurrences and asymptotics of multiple orthogonal polynomials”, Constr. Approx., 30:2 (2009), 175–223 | DOI | MR | Zbl

[11] G. V. Chudnovsky, “Hermite–Padé approximations to exponential functions and elementary estimates of the measure of irrationality of $\pi$”, The Riemann problem, complete integrability and arithmetic applications (Bures-sur-Yvette/New York, 1979/1980), Lecture Notes in Math., 925, Springer-Verlag, Berlin–New York, 1982, 299–322 | DOI | MR | Zbl

[12] W. Van Assche, “Multiple orthogonal polynomials, irrationality and transcedence”, Continued fractions: from analytic number theory to constructive approximation (Columbia, MO, 1998), Contemp. Math., 236, Amer. Math. Soc., Providence, RI, 1999, 325–342 | DOI | MR | Zbl

[13] K. Mahler, “Applications of some formulae by Hermite to the approximation of exponentials and logarithms”, Math. Ann., 168 (1967), 200–227 | DOI | MR | Zbl

[14] Rational approximants for the Euler constant and recurrence relations, Proc. Steklov Inst. Math., 272, suppl. 2 (2011), S138–S190 | DOI | DOI | Zbl

[15] A. I. Aptekarev, H. Stahl, “Asymptotics of Hermite–Padé polynomials”, Progress in approximation theory (Tampa, FL, 1990), Springer Ser. Comput. Math., 19, Springer-Verlag, New York, 1992, 127–167 | DOI | MR | Zbl

[16] A. I. Aptekarev, V. I. Buslaev, A. Martínez-Finkelshtein, S. P. Suetin, “Padé approximants, continued fractions, and orthogonal polynomials”, Russian Math. Surveys, 66:6 (2011), 1049–1131 | DOI | DOI | MR | Zbl

[17] A. I. Aptekarev, A. Kuijlaars, “Hermite–Padé approximations and multiple orthogonal polynomial ensembles”, Russian Math. Surveys, 66:6 (2011), 1133–1199 | DOI | DOI | MR | Zbl

[18] H. Stahl, “Asymptotics for quadratic Hermite–Padé polynomals associated with the exponential function”, Electron. Trans. Numer. Anal., 2002, no. 14, 195–222 | MR | Zbl

[19] G. López Lagomasino, S. Medina Peralta, U. Fidalgo Prieto, “Hermite-Padé approximation for certain systems of meromorphic functions”, Sb. Math., 206:2 (2015), 225–241 | DOI | DOI | MR | Zbl

[20] Ch. Hermite, “Sur la fonction exponentielle”, C. R. Akad. Sci. (Paris), 77 (1873), 18–24, 74–79, 226–233, 285–293 | Zbl

[21] F. Klein, Elementarmathematik vom höheren Standpunkte aus, v. 1, Arithmetik, Algebra, Analysis, 3. Aufl., Springer-Verlag, Berlin, 1924, xii+321 pp. | MR | MR | Zbl | Zbl

[22] A. I. Aptekarev, “Convergence of rational approximations to a set of exponents”, Mosc. Univ. Math. Bull., 36:1 (1981), 81–86 | MR | Zbl

[23] H. Padé, “Mémoire sur les développements en fractions continues de la fonction exponentielle, pouvant servir d'introduction à la théorie des fractions continues algébriques”, Ann. Sci. École Norm. Sup. (3), 16 (1899), 395–426 | MR | Zbl

[24] A. P. Starovoitov, “Approksimatsii Ermita–Pade dlya sistemy funktsii Mittag-Lefflera”, PFMT, 2013, no. 1(14), 81–87 | Zbl

[25] A. P. Starovoitov, “O svoistvakh approksimatsii Ermita–Pade dlya sistemy funktsii Mittag-Lefflera”, Dokl. NAN Belarusi, 57:1 (2013), 5–10

[26] A. P. Starovoitov, “The asymptotic form of the Hermite–Padé approximations for a system of Mittag-Leffler functions”, Russian Math. (Iz. VUZ), 58:9 (2014), 49–56 | DOI | MR | Zbl

[27] A. B. J. Kuijlaars, H. Stahl, W. Van Assche, F. Wielonsky, “Type II Hermite–Padé approximation to the exponential function”, J. Comput. Appl. Math., 207:2 (2007), 227–244 | DOI | MR | Zbl

[28] Ch. Hermite, “Sur la généralisation des fractions continues algébriques”, Ann. Mat. Pura Appl. Ser. 2A, 21 (1893), 289–308 | DOI | Zbl

[29] K. Mahler, “Zur Approximation der Exponentialfunktion und des Logarithmus”, I, J. Reine Angew. Math., 166 (1932), 118–136 ; II, 137–150 | DOI | MR | Zbl | DOI | Zbl

[30] K. Mahler, “Perfect systems”, Compositio Math., 19 (1968), 95–166 | MR | Zbl

[31] E. M. Nikishin, V. N. Sorokin, Rational approximations and orthogonality, Transl. Math. Monogr., 92, Amer. Math. Soc., Providence, RI, 1991, viii+221 pp. | MR | MR | Zbl | Zbl

[32] P. B. Borwein, “Quadratic Hermite–Padé approximation to the exponential function”, Constr. Approx., 2:4 (1986), 291–302 | DOI | MR | Zbl

[33] F. Wielonsky, “Asymptotics of diagonal Hermite–Padé approximants to $e^z$”, J. Approx. Theory, 90:2 (1997), 283–298 | DOI | MR | Zbl

[34] A. P. Starovoitov, “Asimptotika kvadratichnykh approksimatsii Ermita–Pade eksponentsialnykh funktsii”, PFMT, 2014, no. 1(18), 74–80 | Zbl

[35] L. N. Trefethen, “The asymptotic accuracy of rational best approximations to $e^z$ on a disk”, J. Approx. Theory, 40:4 (1984), 380–383 | DOI | MR | Zbl

[36] D. Braess, “On the conjecture of Meinardus on rational approximation of $e^x$. II”, J. Approx. Theory, 40:4 (1984), 375–379 | DOI | MR | Zbl

[37] Yu. V. Sidorov, M. V. Fedoryuk, M. I. Shabunin, Lektsii po teorii funktsii kompleksnogo peremennogo, 3-e izd., Nauka, M., 1989, 478 pp. | MR | Zbl

[38] 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

[39] G. Polia, G. Segë, Zadachi i teoremy iz analiza, v. I, 3-e izd., Nauka, M., 1978, 392 pp.; G. Pólya, G. Szegö, Aufgaben und Lehrsätze aus der Analysis, v. 1, Grundlehren Math. Wiss., 19, 3. bericht. Aufl., Springer-Verlag, Berlin–New York, 1964, xvi+338 pp. ; G. Pólya, G. Szegö, Problems and theorems in analysis, т. 1, Springer Study Edition, Springer-Verlag, New York–Heidelberg–Berlin, 1976, xix+389 с. | MR | Zbl | Zbl

[40] A. B J. Kuijlaars, W. Van Assche, F. Wielonsky, “Quadratic Hermite–Padé approximation to the exponential function: a Riemann–Hilbert approach”, Constr. Approx., 21:3 (2005), 351–412 | DOI | MR | Zbl

[41] G. Szegö, “Über eine Eigenschaft der Exponentialreihe”, Sitzungsber. Berl. Math. Ges., 1924, no. 23, 50–64 | Zbl

[42] E. B. Saff, R. S. Varga, “On the zeros and poles of Padé approximants to $e^z$. II”, Pade and rational approximation. Theory and applications (Univ. South Florida, Tampa, FL, 1976), Academic Press, New York, 1977, 195–213 | DOI | MR | Zbl

[43] A. V. Astafeva, A. P. Starovoitov, “Ekstremalnye svoistva approksimatsii Ermita–Pade eksponentsialnykh funktsii”, Dokl. NAN Belarusi, 58:2 (2014), 32–37