Abstract
Given a formal power seriesf(z)≔∑ ∞ j=0 a j z j for which the quantitya j −1a j +1/a 2 j has a prescribed asymptotic behavior asj→∞, we obtain the asymptotic behavior of poles of rows of the Padé table, and the associated Toeplitz determinants. In particular, we can show for large classes of entire functions of zero, finite, and infinite order (including the Mittag-Leffler functions) and forn=1,2,3,..., that the poles of [m/n](z) approach ∞ with ratea m /a m+1 asm→∞.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. I. Aptekarev (1982):Asymptotics of Hadamard determinants and the convergence of rows of the Padé table for sums of exponentials, Math. USSR-Sb.,41:427–441.
G. A. Baker, Jr. (1975): Essentials of Padé Approximants. New York: Academic Press.
G. A. Baker, Jr., P. Graves-Morris (1977):Convergence of rows of the Padé table. J. Math. Anal. Appl.,57:323–339.
G. A. Baker, Jr., P. Graves-Morris (1981): Padé Approximants, Part I: Basic Theory. In: Encyclopaedia of Mathematics and Its Applications. Reading: Addison-Wesley.
G. A. Baker, Jr., D. S. Lubinsky (1987):Convergence theorems for rows of differential and algebraic Hermite-Padé approximations. J. Comput. Appl. Math. (To appear).
C. Brezinski (1980): Padé Type Approximation and General Orthogonal Polynomials. Basel: Birkhäuser-Verlag.
V. I. Buslaev, A. A. Goncar, S. P. Suetin (1984):On the convergence of subsequences of the mth row of the Padé table. Math. USSR-Sb,48: 535–540.
A. Cuyt (1985):A Montessus de Balloré theorem for multivariate Padé approximants. J. Approx. Theory,43:43–52.
A. Edrei (1978):Angular distribution of the zeros of Padé polynomials. J. Approx. Theory,24:251–265.
A. Edrei (1980):The Padé tables of entire functions, J. Approx. Theory,28:54–82.
A. Edrei, E. B. Saff, R. S. Varga (1983): Zeros of Sections of Power Series. Lecture Notes in Mathematics, Vol. 1002. Berlin: Springer-Verlag.
A. P. Golub (1984):Padé approximation of the Mittag-Leffter function. In: The Theory of Approximation of Functions and Its Applications, Akad. Nauk Ukrain. SSR.,136:52–59 (in Russian: Mathematical Reviews, Vol. 86f: 30037).
P. R. Graves-Morris, E. B. Saff (1984):Vector valued rational interpolants. In: Rational Approximation and Interpolation (P. R. Graves-Morris, E. B. Saff, R. S. Varga, eds.). Lecture Notes in Mathematics, Vol. 1105. Berlin: Springer-Verlag, pp. 227–242.
A. L. Levin, D. S. Lubinsky (1986):Best rational approximation of entire functions whose Maclaurin series coefficients decrease rapidly and smoothly. Trans. Amer. Math. Soc.,293:533–546.
G. Lopez, V. V. Vavilov (1984):Survey on recent advances in inverse problems of Padé approximation theory. In: Rational Approximation and Interpolation. (P. R. Graves-Morris, E. B. Saff, R. S. Varga, eds.). Lecture Notes in Mathematics, Vol. 1105. Berlin: Springer-Verlag, pp. 11–26.
D. S. Lubinsky (1980):On nondiagonal Padé approximants. J. Math. Anal. Appl.,78:405–428.
D. S. Lubinsky (1985):Padé tables of a class of entire functions. Proc. Amer. Math. Soc.,94:399–405.
D. S. Lubinsky (1985):Padé tables of entire functions of very slow and smooth growth. Constr. Approx.,1:349–358.
D. S. Lubinsky (1987):Asymptotics of Toeplitz determinants and rows of the Padé table. In: Approximation Theory V. (To appear).
D. S. Lubinsky, E. B. Saff (1987):Padé approximants of partial theta functions and the Rogers-Szegö polynomials. In: Approximation Theory V. (To appear).
D. S. Lubinsky, E. B. Saff (1987):Convergence of Padé approximants of partial theta functions and the Rogers-Szegö polynomials. Constr. Approx. (To appear).
F. W. J. Olver (1974): Asymptotics and Special Functions. New York: Academic Press.
E. B. Saff (1972):An extension of Montessus de Ballore's theorem on the convergence of interpolating rational functions. J. Approx. Theory,6:63–67.
E. B. Saff (1969):On the row convergence of the Walsh array for meromorphic functions. Trans. Amer. Math. Soc.,146:241–257.
H. Stahl (1976): Beiträge zum Problem der Konvergenz von Padé approximierenden. Ph.D. Thesis. Technical University of Berlin.
H. Wallin (1900):Potential theory and approximation of analytic functions by rational interpolation. In: Proc. Coll. Complex Anal., Joensuu, Finland. Lecture Notes in Mathematics, Vol. 888. Berlin: Springer-Verlag, pp. 371–383.
H. Wallin, (1984):Convergence and divergence of multipoint Padé approximants of meromorphic functions. In: Rational Approximation and Interpolation, (P. R. Graves-Morris, E. B. Saff, R. S. Varga, eds.). Lecture Notes in Mathematics, Vol. 1105. Berlin: Springer-Verlag, pp. 272–284.
R. Wilson (1928):Divergent continued fractions and nonpolar singularities. Proc. London Math. Soc.,28:128–144.
Author information
Authors and Affiliations
Additional information
Communicated by Edward B. Saff.
Rights and permissions
About this article
Cite this article
Lubinsky, D.S. Uniform convergence of rows of the Padé table for functions with smooth Maclaurin series coefficients. Constr. Approx 3, 307–330 (1987). https://doi.org/10.1007/BF01890573
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01890573