Asymptotic Properties of Orthogonal Polynomials with Respect to a Non-discrete Jacobi-Sobolev Inner Product

摘要

Let {Q(n)((alpha,beta)) (x)}(n=0)(infinity) denote the sequence of polynomials orthogonal with respect to the non-discrete Sobolev inner product < f, g > = integral(1)(-1)f(x)g(x)d mu(alpha,beta)(x) + lambda integral(1)(-1)f'(x)g'(x)dv(alpha,beta)(x) where lambda > 0 and d mu alpha, beta( x) = (x-a)(1-x)(alpha-1)(1+ x)(beta-1)dx, d nu(alpha,beta)(x) = (1-x)(alpha)(1+ x)(beta)dx with a -1, alpha, beta > 0. Their inner strong asymptotics on (-1, 1), a Mehler-Heine type formula as well as some estimates of the Sobolev norms of Q(n)((alpha, beta)) n are obtained.