# Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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## The rate of convergence of Chebyshev polynomials for functions which have asymptotic power series about one endpointHTML articles powered by AMS MathViewer

by John P. Boyd
Math. Comp. 37 (1981), 189-195 Request permission

## Abstract:

The theorem proved here extends Chebyshev theory into what has previously been no man’s land: functions which have an infinite number of bounded derivatives on the expansion interval [a, b] but which are singular at one endpoint. The Chebyshev series in $1/x$ for all the familiar special functions fall into this category, so this class of functions is very important indeed. In words, the theorem shows that the more slowly the asymptotic power series about the singular point converges, the slower the convergence of the corresponding Chebyshev series must be. More formally, if $f(x)$, analytic on [a, b), is singular at $x = b$ in such a way that it has an asymptotic power series $f(x) \sim \Sigma {a_n}{(x - b)^n}$ about that endpoint, then, if $\overline {\lim \limits _{n \to \infty } } \frac {{\log |{a_n}|}}{{n\log n}} = r,$ it is proved that the coefficients of the convergent Chebyshev polynomial series on [a, b], $f(x) \sim \Sigma {b_n}{T_n}(y)$ where $y = 2[x - 0.5(b + a)]/(b - a)$, satisfy the inequality $\overline {\lim \limits _{n \to \infty } } \frac {{\log |(\log |{b_n}|)|}}{{\log n}} \leqslant \frac {2}{{r + 2}}$.
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