The rate of convergence of Chebyshev polynomials for functions which have asymptotic power series about one endpoint

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

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

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

$$\overline {\mathop {\lim }\limits_{n \to \infty } } \frac{{\log \vert(\log \vert{b_n}\vert)\vert}}{{\log n}} \leqslant \frac{2}{{r + 2}}$$

.