Symmetrical Orthogonal Polynomials for Sobolev-Type Inner Products

Abstract

In this paper, families of symmetric orthogonal polynomials (Qn) with respect to the Sobolev-type inner product, 〈f, g〉=∫Ifgdμ+Σrj=0Mjƒ(j)(0) g(j)(0) where I is a symmetric interval and μ is a symmetric positive Borel measure with infinite support on I and whose moments are all finite, are considered. If Q2n(x)=Un(x2) and Q2n+1(x)=xVn(x2), we deduce that Un and Vn are Sobolev-type orthogonal polynomials and, in several particular cases, standard orthogonal polynomials. We study the zeros of Qn showing that, in some cases, Qn has two complex conjugate zeros; moreover a partial result about separation of the zeros is given. We also discuss the symmetrization problem for this kind of inner products. Finally, some Sobolev-type inner products with two symmetric mass points are considered.