Sobolev orthogonal polynomials and second order differential equation II

摘要

We obtain necessary and sufficient conditions for a sequence of polynomials to be orthogonal relative to the Sobolev pseudo-inner product $phi (cdot , cdot ) $ defined by $$phi (p, q)=sum_{k=0}^N int_{Bbb R} p^{(k)}q^{(k)},dmu_k $$ where each $dmu_k$ is a signed Borel measure and to satisfy a second-orderdifferential equation of the form $$ ell_2(x)y''(x)+ell_1(x)y'(x)=lambda_ny(x). $$ Using these results, we classify all such Sobolev orthogonal polynomialsin case of $N=2$.This classification generalizes the well known Bochner classification of theclassical orthogonal polynomials corresponding to the case $N=0$ and the classification by K. H. Kwon and L. L. Littlejohn for the case $N=1$.