Orthogonality of linear combinations of two sequences of orthogonal polynomials

Abstract

We find necessary and sufficient conditions for some linear combinations of two sequences of monic polynomials orthogonal with respect to linear functionals u, v, respectively, be again orthogonal. The situation of convex linear combinations is analyzed more carefully. Some kind of perturbations in the parameters of the three-term recurrence relation appear in a natural way.

Finally, we compare our results with a previous one ([1]) for orthogonal polynomials on the unit circle.

This is a joint work with K. H. Kwon and J. H. Lee (KAIST, Taejon, South Korea).

Reference

1. A. Cachafeiro and F. Marcellan, On convex linear combinations of sequences of monic polynomials, Proc. Amer. Math. Soc. 126 (1998), 2323-2331.