An optimal property of Chebyshev expansions

Abstract

Chebyshev polynomials are extremely popular in numerical analysis. One of their virtues is that expansions of functions in series of Chebyshev polynomials are thought to converge more rapidly than expansions in series of other orthogonal polynomials, and some supporting asymptotic evidence for this belief is presented in Lanczos [2]. Our purpose here is to demonstrate that for a certain restricted class of functions, the truncated Chebyshev expansion is best in some fairly large class of Jacobi expansions, and, thus, to provide further solid foundation for the Chebyshev faith. The remainder of the Introduction is devoted to presenting notation and setting the stage. In Section 1, we make precise the sense in which Chebyshev expansions are best, while Section 2 is given over to various counter-examples to the results in Section 1. Let Pp, s’(x) be the Jacobi polynomials with CL, ,l3 > -1 (that is, the orthogonal polynomials on I: [-1, l] with respect to the weight function w(oz,/~;x) = (1 x)a(l +x)“), normalized in the usual fashion (cf. Szegii [4], p. 58)). For each (cc, /I), (y, 8) we have