Jacobi–Sobolev orthogonal polynomials: Asymptotics and a Cohen type inequality

Abstract

Let dμα,β(x)=(1−x)α(1+x)βdx,α,β>−1, be the Jacobi measure supported on the interval [−1,1]. Let us introduce the Sobolev inner product 〈f,g〉S=∑j=0Nλj∫−11f(j)(x)g(j)(x)dμα,β(x), where λj≥0 for 0≤j≤N−1 and λN>0. In this paper we obtain some asymptotic results for the sequence of orthogonal polynomials with respect to the above Sobolev inner product. Furthermore, we prove a Cohen type inequality for Fourier expansions in terms of such polynomials.