On Recurrence Relations for Sobolev Orthogonal Polynomials

Abstract

This paper discusses recurrence relations for sequences of polynomials which are orthogonal with respect to the Sobolev inner product defined on the set of polynomials $\mathcal{P}$ by \[ (p,q)w = \sum_{k = 0}^N {\int_\mathbb{R} {p^{(k)} (x)\bar q^{(k)} (x)d\mu _k (x)\quad (p,q \in \mathcal{P})} } \] for some integer $N \geq 1$, where each $\mu _k $, $0 \leq k \leq N$, is a positive Borel measure. It is proven that there exists a real-valued polynomial $h:\mathbb{R} \to \mathbb{R}$ satisfying \[ ( * )\qquad (hp,q)_W = (p,hq)_W\quad(p,q \in \mathcal{P})\] if and only if each of the measures $\mu _k $, $1 \leq k \leq N$, is purely atomic with a finite number of mass points. In addition it is proven that $R_j $, the set of real roots of ${{d^j h} / {dx^j }}$, $(1 \leq j \leq N)$, is nonempty and that $(\mu _k ) \subset \cap _{i = 1}^k R_i $. It is also shown that if h satisfies the condition $(*)$, then the polynomials orthogonal with respect to the inner product $( \cdot , \cdot )W$ will satisfy a recurrence relation of order $2m + 1$, where $m = \deg (h)$. Furthermore, an algorithm is given to construct a polynomial H of minimal positive degree forr which the above properties hold. Several examples will be discussed to illustrate the theory. Lastly it is shown, under certain circumstances, when these orthogonal polynomials will satisfy second-order linear differential equations.