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A Chebyshev polynomial rate-of-convergence theorem for Stieltjes functions

Author: John P. Boyd
Journal: Math. Comp. 39 (1982), 201-206
MSC: Primary 41A25
MathSciNet review: 658224
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Abstract: The theorem proved here extends the author's previous work on Chebyshev series [4] by showing that if $ f(x)$ is a member of the class of so-called "Stieltjes functions" whose asymptotic power series $ \Sigma {a_n}{x^n}$ about $ x = 0$ is such that

$\displaystyle \overline {\mathop {\lim }\limits_{n \to \infty } } \frac{{\log \vert{a_n}\vert}}{{n\log n}} = r,$

then the coefficients of the series of shifted Chebyshev polynomials on $ x \in [0,a],\Sigma {b_n}T_n^\ast(x/a)$, satisfy the inequality

$\displaystyle \frac{2}{{r + 2}} \geqslant \overline {\mathop {\lim }\limits_{n ... ...c{{\log \vert(\log \vert{b_n}\vert)\vert}}{{\log n}} \geqslant 1 - \frac{r}{2}.$

There is an intriguing relationship between this theorem and a similar rate-of-convergence theorem for Padé approximants of Stieltjes functions which is discussed below.

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Keywords: Chebyshev polynomial series
Article copyright: © Copyright 1982 American Mathematical Society