Padé tables of a class of entire functions
Author:
D. S. Lubinsky
Journal:
Proc. Amer. Math. Soc. 94 (1985), 399405
MSC:
Primary 30E10; Secondary 41A21
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 BakerGammelWills Conjecture is of the second category.
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 [1]
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 [2]
 G. A. Baker, Jr., Essentials of Padé approximants, Academic Press, New York, 1975. MR 0454459 (56:12710)
 [3]
 , Convergence of Padé approximants using the solution of linear functional equations, J. Math. Phys. 16 (1975), 813822. MR 0366279 (51:2527)
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 P. B. Borwein, The usual behaviour of rational approximations, Canad. Math. Bull. 26 (1983), 317323. MR 703403 (84f:41019)
 [5]
 V. I. Buslaev, A. A. Gončar and S. P. Suetin, Convergence of subsequences of the th row of a Padé table, Mat. Sb. 120 (1983), 540545. (Russian) MR 695959 (85c:41026)
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 A. Edrei, The Padé table of meromorphic functions of small order with negative zeros and positive poles, Rocky Mountain J. Math. 4 (1974), 175180. MR 0340610 (49:5362)
 [7]
 , The Padé table of functions having a finite number of essential singularities, Pacific J. Math. 56 (1975), 429453. MR 0393498 (52:14308)
 [8]
 A. A. Gončar and K. N. Lungu, Poles of diagonal Padé approximants and the analytic continuation of functions, Math. USSR.Sb. 39 (1981), 255266.
 [9]
 E. Hendriksen and H. Van Rossum, Moment methods in Padé approximation, J. Approx. Theory 35 (1983), 250263. MR 663671 (83i:41020)
 [10]
 D. S. Lubinsky, Diagonal Padé approximants and capacity, J. Math. Anal. Appl. 78 (1980), 5867. MR 595764 (82e:30050)
 [11]
 , Divergence of complex rational approximations, Pacific J. Math. 108 (1983), 141153. MR 709706 (84i:41024)
 [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), 333345. MR 697078 (84d:41030)
 [13]
 A. M. Ostrowski, Note on bounds for determinants with dominant principal diagonal, Proc. Amer. Math. Soc. 3 (1952), 2630. MR 0052380 (14:611c)
 [14]
 Ch. Pommerenke, Padé approximants and convergence in capacity, J. Math. Anal. Appl. 41 (1973), 775780. MR 0328090 (48:6432)
 [15]
 E. A. Rahmanov, On the convergence of Padé approximants in classes of holomorphic functions, Math. USSRSb. 40 (1981), 149155.
 [16]
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 [17]
 , Potential theory and approximation of analytic functions by rational interpolation, Proc. Complex Analysis Conf. at Joensuu (I. Laine, O. Lehto, T. Sorvali, Editors), Lecture Notes in Math., Vol. 747, Springer, Berlin, 1979, pp. 434450. MR 553073 (81i:30069)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198507878819
PII:
S 00029939(1985)07878819
Keywords:
Padé approximation,
entire functions,
uniform convergence
Article copyright:
© Copyright 1985
American Mathematical Society
