On an inverse problem for a linear combination of orthogonal polynomials

Abstract

This paper deals with the analysis of the orthogonality of a monic polynomial sequence defined as a linear combination of a sequence of monic orthogonal polynomials with

where for . Moreover, we obtain the relation between the corresponding linear functionals as well as an explicit expression for the sequence of monic orthogonal polynomials . We obtain the connection between the Jacobi matrices associated with and , respectively, by using an LU factorization. Some special cases of the above type relation are analysed.

Keywords::

• orthogonal polynomials
• linear functionals
• Jacobi matrices
• inverse problems

Keywords::

• 33C45
• 42C05

Acknowledgements

First of all, the authors like to thank the referees for their careful revision of the manuscript. Their comments and suggestions have substantially improved the presentation of the manuscript. The work of the first author (FM) has been supported by Ministerio de Economía y Competitividad of Spain, grant MTM 2012-36732-C03-01. This paper was completed during a stay of the second author (SV) in the Department of Mathematics of Universidad Carlos III de Madrid in the spring semester of the academic year 2012–2013. He acknowledges the kind reception there.

Notes

^1. Email: [email protected]