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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Best rational approximations of entire functions whose Maclaurin series coefficients decrease rapidly and smoothly

Authors: A. L. Levin and D. S. Lubinsky
Journal: Trans. Amer. Math. Soc. 293 (1986), 533-545
MSC: Primary 30E10; Secondary 41A20
MathSciNet review: 816308
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Abstract: Let $ f = \Sigma _{j = 0}^\infty {a_j}{z^j}$ be an entire function which satisfies

$\displaystyle \vert{a_{j - 1}}a{ _{j + 1}}/a_j^2\vert \leqslant {\rho ^2},\qquad j = 1,2,3, \ldots ,$

where $ 0 < \rho < {\rho _0}$ and $ {\rho _0} = 0.4559 \ldots $ is the positive root of the equation $ 2\Sigma _{j = 1}^\infty {\rho ^{{j^2}}} = 1$. Let $ r > 0$ be fixed. Let $ {W_{LM}}$ denote the rational function of type $ (L,M)$ of best approximation to $ f$ in the uniform norm on $ \vert z\vert \leqslant r$. We show that for any sequence of nonnegative integers $ \{ {M_L}\} _{L = 1}^\infty $ that satisfies $ {M_L} \leqslant 10L,\,L = 1,2,3, \ldots $, the rational approximations $ {W_{L{M_L}}}$ converge to $ f$ throughout $ {\mathbf{C}}$ as $ L \to \infty $. In particular, convergence takes place for the diagonal sequence and for the row sequences of the Walsh array for $ f$.

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Keywords: Padé table, best rational approximation, Walsh array
Article copyright: © Copyright 1986 American Mathematical Society