An 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 $L^2({\mathbb{R}}^n)$ under the Hermite semigroup. We also obtain some new orthogonality relations for complexified Hermite functions.

15 pages