A note on the Geronimus transformation and Sobolev orthogonal polynomials

TL;DR

This paper explores the Geronimus transformation within the context of symmetric bilinear forms and demonstrates how double transformations result in non-diagonal Sobolev inner products, advancing understanding of orthogonal polynomial transformations.

Contribution

It introduces a new perspective on the Geronimus transformation and connects it to Sobolev orthogonal polynomials through double transformations.

Findings

Geronimus transformation recast in symmetric bilinear form framework

Double Geronimus transformations produce non-diagonal Sobolev inner products

Provides a new approach to orthogonal polynomial transformations

Abstract

In this note we recast the Geronimus transformation in the framework of polynomials orthogonal with respect to symmetric bilinear forms. We also show that the double Geronimus transformations lead to non-diagonal Sobolev type inner products.