An analogue of Gutzmer's formula for Hermite expansions

TL;DR

This paper develops a Hermite expansion analogue of Gutzmer's formula, providing new proofs and orthogonality relations that deepen understanding of Hermite semigroup images and complexified Hermite functions.

Contribution

It introduces a novel analogue of Gutzmer's formula for Hermite expansions, offering new proofs and orthogonality relations in the context of Hermite analysis.

Findings

New proof of the Hermite semigroup image characterization

Orthogonality relations for complexified Hermite functions

Analogue of Gutzmer's formula for Hermite expansions

Abstract

We prove an analogue of Gutzmer's formula for Hermite expansions. As a consequence we obtain a new proof of a characterisation of the image of $ L^2(\R^n) $ under the Hermite semigroup. We also obtain some new orthogonality relations for complexified Hermite functions.