h-Transforms and Orthogonal Polynomials

TL;DR

This paper explores classical h-transforms within the framework of symmetric diffusion operators that possess orthogonal polynomials as their spectral basis, providing explicit examples and a unifying mechanism.

Contribution

It introduces a general mechanism linking h-transforms to symmetric diffusion operators with orthogonal polynomial spectra, with specific classical examples.

Findings

Identifies explicit classical h-transforms as special cases.

Connects h-transforms to spectral decompositions of diffusion operators.

Provides a unified framework for understanding these transforms.

Abstract

We describe some examples of classical and explicit h-transforms as particular cases of a general mechanism, which is related to the existence of symmetric diffusion operators having orthogonal polynomials as spectral decomposition.