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 l/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 /(*), analytic on [a, b), is singular at x — b in such a way that it has an asymptotic power series/(x) ~ 2 a„(x — by about that endpoint, then, if islogkl lim —;-« r, «-.oc nlogn it is proved that the coefficients of the convergent Chebyshev polynomial series on [a, b], f(x) ~ 2 b„T„(y) where.y = 2[x 0.5(b + a)]/(b a), satisfy the inequality — log|(log|¿„|)| ^ 2 urn -j-< ——r. »-.oo log n r + 2 It is well known that if a function f(x) is singular about a point, let us say x = 1 to be definite, then the power series of f(x) about that point is at most asymptotic and must diverge. The corresponding series of Chebyshev polynomials on [-1, 1], which includes the singularity as an endpoint, is much more robust. If the singularity is weak, in the sense that all (left) derivatives of f(x) are bounded at x = 1, then it can be proven by a simple integration-by-parts argument [1] that the Chebyshev series converges absolutely and exponentially on the interval. Indeed, such series, usually in the form of series of shifted Chebyshev polynomials in l/x, are enormously useful in approximation theory for representing the large x behavior of Bessel functions and many other common transcendentals. Luke [2] gives extensive tables of such approximations. Although a Chebyshev series, whose coefficients are 0(e~*" ), technically possesses the property of exponential or "infinite order" convergence for any positive ß, the precise value of ß is obviously of great practical importance. The integrationby-parts argument implies only that ß > 0. When a function has a singularity which is not on [-1, 1], it is known that bn ~ ae_x" for some a and a which depend Received March 21, 1980; revised August 21, 1980 and October 29, 1980. AMS (MOS) subject classifications (1970). Primary 42A56; Secondary 33A65, 41A10.