Hyperfunctional Weights for Orthogonal Polynomials

Abstract

The Chebychev polynomials associated to any given moments μn ∞ 0 are formally orthogonal with respect to the formal δ-series w(x)=∑ 0 ∞ (−1) n μ n δ (n) (x)/n!. We show that this formal weight can be a true hyperfunctional weight if its Fourier transform is a slowly increasing holomorphic function in some tubular neighborhood of the real line. It provides a unifying treatment of real and complex orthogonality of Chebychev polynomials including all classical examples and characterizes Chebychev polynomials having Bessel type orthogonality.