Companion linear functionals and Sobolev inner products

Abstract

The present paper deals with the solution of an inverse problem in the theory of orthogonal polynomials. It was motivated by a characterization result concerning sequences of polynomials orthogonal with respect to a Sobolev inner product when they can be recursively generated in terms of orthogonal polynomial sequences associated with the measure involved in the standard component. More precisely, we obtain the set of pairs of quasi–definite linear functionals such that their corresponding sequences of monic orthogonal polynomials {Pn} and {Rn} are related by a differential expression $$ \frac{R'_{n+1}(x)}{n+1}+b_n\frac{R'_n(x)}{n}=P_n(x)+a_nP_{n-1}(x) \tag 2$$ where $ b_n\ne0$ for every n ∈ N.