A Chebyshev Polynomial Rate-of-Convergence Theorem for Stieltjes Functions

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 2 anx" about x = 0 is such that ttlogjaj _ hm —;-= r, «log n then the coefficients of the series of shifted Chebyshev polynomials on x e [0, a],2b„Tf(x/a), satisfy the inequality 2 . m log | (log |M) | >x r r + 2 log « 2 ' «—00 ° 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. A Stieltjes function is defined by the integral where the weight function p(t) is such that (2) p(t)>0, re[0,oo], and all the moment integrals (3) an= t"p(t)dt exist for all nonnegative integers n [3]. Although special, this class of functions plays a major role in nuclear, atomic, and elementary particle theory, in critical phenomenon, and in a variety of other fields as shown in the books by Baker [2], Graves-Morris [8], [9], and Cabannes [6]. One reason for the great attention given to the Stieltjes functions is that one can prove the convergence of the Padé approximants formed from their power series. The purpose of this brief note is to show that one can prove a rate-of-convergence theorem for the Chebyshev polynomial series of these functions which is similar to—and intriguingly related to—that already known for Padé approximants. By expanding (1 + xt)~x in (1), using the binomial theorem, and then integrating term-by-term, one can show that the coefficients of the power series expansion of Received March 17, 1981; revised November 18, 1981. 1980 Mathematics Subject Classification. Primary 42A56; Secondary 33A65, 41A10.