Zeros of Orthogonal Polynomials in a Discrete Sobolev Space with a SymmetricInner Product

摘要

Recently several authors studied orthogonal polynomials in a discrete Sobolevspace, where the inner product is of the form = integral from a to b of f(x) g(x) d alpha(x) + Summation k = 1 to k = r of N(sub k)(c)g(supk)(c). In two preceeding papers, the authors considered orthogonal polynomials with respect to the following inner product which is symmetric with respect to the origin: = integral from -1 to +1 of f(x)g(x) w(x) dx + M (f(1)g(1) + f(-1)g(-1)) + N(f'(1)g'(1) + f'(-1)g'(-1)), where M = or > 0, N = or > 0 and w(x) = (1-x(squared))(sup alpha) with alpha > -1. The authors computed the polynomials and they deduced some properties concerning the zeros of these polynomials. In the present paper the weight function (1-x(squared))(sup alpha) will be replaced by a more general non-negative symmetric weight function w on (-1,1) such that w(-x) = w(x), the integrals c(sub i) = integral from -1 to +1 of (1-x(squared))(sup i) w(x) dx exist and are positive for all i epsilon (0, 1, 2, ...) and the distribution w(x) dx has infinitely many points of increase.