Padé tables of a class of entire functions

Author:
D. S. Lubinsky

Journal:
Proc. Amer. Math. Soc. **94** (1985), 399-405

MSC:
Primary 30E10; Secondary 41A21

DOI:
https://doi.org/10.1090/S0002-9939-1985-0787881-9

MathSciNet review:
787881

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Abstract: It is shown that if the Maclaurin series coefficients of an entire function satisfy a certain explicit condition, then there exists a sequence of integers such that locally uniformly in as , for all nonnegative integer sequences . In particular, this condition is satisfied if the approach 0 fast enough, or if a subsequence of the behaves irregularly in a certain sense. Further, the functions satisfying this condition are dense in the space of entire functions with the topology of locally uniform convergence. Consequently, the set of entire functions satisfying the Baker-Gammel-Wills Conjecture is of the second category.

**[1]**Robert J. Arms and Albert Edrei,*The Padé tables and continued fractions generated by totally positive sequences*, Mathematical Essays Dedicated to A. J. Macintyre, Ohio Univ. Press, Athens, Ohio, 1970, pp. 1–21. MR**0276452****[2]**George A. Baker Jr.,*Essentials of Padé approximants*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR**0454459****[3]**George A. Baker Jr.,*Convergence of Padé approximants using the solution of linear functional equations*, J. Mathematical Phys.**16**(1975), 813–822. MR**0366279**, https://doi.org/10.1063/1.522610**[4]**Peter B. Borwein,*The usual behaviour of rational approximations*, Canad. Math. Bull.**26**(1983), no. 3, 317–323. MR**703403**, https://doi.org/10.4153/CMB-1983-051-1**[5]**V. I. Buslaev, A. A. Gonchar, and S. P. Suetin,*Convergence of subsequences of the 𝑚th row of a Padé table*, Mat. Sb. (N.S.)**120(162)**(1983), no. 4, 540–545 (Russian). MR**695959****[6]**Albert Edrei,*The Padé table of meromorphic functions of small order with negative zeros and positive poles*, Proceedings of the International Conference on Padé Approximants, Continued Fractions and Related Topics (Univ. Colorado, Boulder, Colo., 1972; dedicated to the memory of H. S. Wall), 1974, pp. 175–180. MR**0340610**, https://doi.org/10.1216/RMJ-1974-4-2-175**[7]**Albert Edrei,*The Padé table of functions having a finite number of essential singularities*, Pacific J. Math.**56**(1975), no. 2, 429–453. MR**0393498****[8]**A. A. Gončar and K. N. Lungu,*Poles of diagonal Padé approximants and the analytic continuation of functions*, Math. USSR.-Sb.**39**(1981), 255-266.**[9]**E. Hendriksen and H. van Rossum,*Moment methods in Padé approximation*, J. Approx. Theory**35**(1982), no. 3, 250–263. MR**663671**, https://doi.org/10.1016/0021-9045(82)90007-7**[10]**D. S. Lubinsky,*Diagonal Padé approximants and capacity*, J. Math. Anal. Appl.**78**(1980), no. 1, 58–67. MR**595764**, https://doi.org/10.1016/0022-247X(80)90210-3**[11]**D. S. Lubinsky,*Divergence of complex rational approximations*, Pacific J. Math.**108**(1983), no. 1, 141–153. MR**709706****[12]**D. S. Lubinsky and A. Sidi,*Convergence of linear and nonlinear Padé approximants from series of orthogonal polynomials*, Trans. Amer. Math. Soc.**278**(1983), no. 1, 333–345. MR**697078**, https://doi.org/10.1090/S0002-9947-1983-0697078-1**[13]**A. M. Ostrowski,*Note on bounds for determinants with dominant principal diagonal*, Proc. Amer. Math. Soc.**3**(1952), 26–30. MR**0052380**, https://doi.org/10.1090/S0002-9939-1952-0052380-7**[14]**Ch. Pommerenke,*Padé approximants and convergence in capacity*, J. Math. Anal. Appl.**41**(1973), 775–780. MR**0328090**, https://doi.org/10.1016/0022-247X(73)90248-5**[15]**E. A. Rahmanov,*On the convergence of Padé approximants in classes of holomorphic functions*, Math. USSR-Sb.**40**(1981), 149-155.**[16]**Hans Wallin,*The convergence of Padé approximants and the size of the power series coefficients*, Applicable Anal.**4**(1974), no. 3, 235–251. MR**0393445**, https://doi.org/10.1080/00036817408839094**[17]**Hans Wallin,*Potential theory and approximation of analytic functions by rational interpolation*, Complex analysis Joensuu 1978 (Proc. Colloq., Univ. Joensuu, Joensuu, 1978), Lecture Notes in Math., vol. 747, Springer, Berlin, 1979, pp. 434–450. MR**553073**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1985-0787881-9

Keywords:
Padé approximation,
entire functions,
uniform convergence

Article copyright:
© Copyright 1985
American Mathematical Society