Best rational approximations of entire functions whose Maclaurin series coefficients decrease rapidly and smoothly
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- by A. L. Levin and D. S. Lubinsky PDF
- Trans. Amer. Math. Soc. 293 (1986), 533-545 Request permission
Abstract:
Let $f = \Sigma _{j = 0}^\infty {a_j}{z^j}$ be an entire function which satisfies \[ |{a_{j - 1}}a{ _{j + 1}}/a_j^2| \leqslant {\rho ^2},\qquad j = 1,2,3, \ldots ,\] 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 $|z| \leqslant r$. 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
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Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 293 (1986), 533-545
- MSC: Primary 30E10; Secondary 41A20
- DOI: https://doi.org/10.1090/S0002-9947-1986-0816308-9
- MathSciNet review: 816308