SYMMETRICAL ORTHOGONAL POLYNOMIALS FOR SOBOLEV-TYPE INNER PRODUCTS

[摘要] In this paper, families of symmetric orthogonal polynomials (e,) with respect to the Sobolev-type inner product, =integral(r)fg d mu+Sigma(jm0)(r)M(j)f((1))(0) g((1))(0) where I is a symmetric interval and mu is a symmetric positive Borel measure with infinite support on I and whose moments are all finite, are considered. If Q(2n)(x)=U-n(x(2)) and Q(2n+1)(x)=xV(n)(x(2)), we deduce that U-n and V-n are Sobolev-type orthogonal polynomials and, in several particular cases, standard orthogonal polynomials. We study the zeros of Q(n) showing that, in some cases, Q(n) 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. (C) 1994 Academic Press, Inc.

[发布日期] 1994-06-01