Sobolev orthogonal polynomials and second order differential equation II

Bull. Korean Math. Soc. 1996 Vol. 33, No. 1, 135-170

K. H. Kwon, D. W. Lee and L. L. Littlejohn KAIST, KAIST and Utah state university

Abstract : 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 $$\phi (p, q)=\sum_{k=0}^N \int_{\Bbb R} p^{(k)}q^{(k)}\,d\mu_k $$ where each $d\mu_k$ 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 $N=1$.