On Orthogonal Polynomials of Sobolev Type: Algebraic Properties and Zeros

Abstract

In this paper the inner product $\langle {f,g} \rangle = \int_I {fg\,d\mu } + Mf(c)g(c) + Nf'(c)g'(c)$ is considered, where $\mu $ is a positive measure on the interval I, $c \in {\bf R}$ and M, $N \geqq 0$. General algebraic properties of the orthogonal polynomials associated with $\langle { \cdot , \cdot } \rangle $ as well as the zeros and their location are studied. In particular, the case of a symmetric measure $\mu $ is analyzed. Finally, a second-order linear differential equation and two applications are given.