Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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$.

References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 30E10, 41A20

Retrieve articles in all journals with MSC: 30E10, 41A20


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1986-0816308-9
PII: S 0002-9947(1986)0816308-9
Keywords: Padé table, best rational approximation, Walsh array
Article copyright: © Copyright 1986 American Mathematical Society