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

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 \[ \overline {\lim \limits _{n \to \infty } } \frac {{\log |{a_n}|}}{{n\log n}} = r,\] 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 \[ \frac {2}{{r + 2}} \geqslant \overline {\lim \limits _{n \to \infty } } \frac {{\log |(\log |{b_n}|)|}}{{\log n}} \geqslant 1 - \frac {r}{2}.\] 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.