Gutzmer’s formula and Poisson integrals on the Heisenberg group

Abstract

In 1978 M. Lassalle obtained an analogue of the Laurent series for holomorphic functions on the complexification of a compact symmetric space and proved a Plancherel type formula for such functions. In 2002 J. Faraut established such a formula, which he calls Gutzmer's formula, for all noncompact Riemannian symmetric spaces. This was immediately put into use by B. Krotz, G. Olafsson and R. Stanton to characterise the image of the heat kernel transform. In this article we prove an analogue of Gutzmer's formula for the Heisenberg motion group and use it to characterise Poisson integrals associated to the sublaplacian. We also use the Gutzmer's formula to study twisted Bergman spaces.

Toeplitz Operators with Special Symbols on Segal–Bargmann Spaces

“…e (2k+n)2t . This has been proved in [22], see Lemma 6.3. The following extension of this result is needed for the study of Toeplitz operators.…”

“…Moreover, the norm of F in B * t (C 2n ) is the same as the norm of f in L 2 (R 2n ). Another proof of this was found in [22] which is based on the following Gutzmer's formula for the special Hermite expansion. Recall that ϕ k (x, u) = ϕ k (x + iu) are the Laguerre functions of type (n − 1) introduced earlier.…”

Segal–Bargmann transform and Paley–Wiener theorems on Heisenberg motion groups

“…As in the proof of Theorem 5.1 of [14], for any fixed (y, v) with |y| 2 + |v| 2 ≤ r 2 < which certainly converges if r < p 2 < q 2 . Moreover, using the fact that (f π ij ) λ 1 ≤ f 1 and f λ is compactly supported as a function of λ, we can conclude that…”

A Paley–Wiener theorem for some eigenfunction expansions

“…Our first result is the analogue of Theorem 4.1. The proof of this theorem is similar to that of Theorem 4.1 and is based on the following Gutzmer's formula for special Hermite expansions, see [13]. Theorem 4.6.…”