Geronimus transformation and Sobolev-type orthogonal polynomials

TL;DR

This paper explores how iterated Geronimus transformations generate Sobolev-type orthogonal polynomials, establishing explicit recurrence relations, connection formulas, and asymptotic behaviors, thus linking spectral transformations with Sobolev orthogonality.

Contribution

It provides a unified framework connecting Geronimus transformations with Sobolev orthogonal polynomials, including explicit recurrence relations and asymptotic analysis.

Findings

Derived explicit three-term and five-term recurrence relations.

Established connection formulas via Christoffel-Darboux kernels.

Proved asymptotic ratios of derivatives and norms converge to explicit constants.

Abstract

Iterated Geronimus transformations generate Sobolev-type orthogonal polynomials from classical families. We establish a direct equivalence between a Sobolev inner product involving point evaluation and the first derivative at a point a outside the support of the original measure and two successive Geronimus transformations. Explicit three-term and five-term recurrence relations are derived for the resulting polynomials, revealing their algebraic structure. Connection formulas linking the Sobolev-type polynomials Q_n^{M,N}(x) with both the original and the transformed Geronimus polynomials are obtained via Christoffel-Darboux kernels and determinantal representations. In the Jacobi case, asymptotic analysis shows that ratios of derivatives and norms converge to explicit constants independent of the parameters M and N. These results provide a unified framework connecting spectral transformations with Sobolev orthogonality.