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

Abstract

<jats:title>Abstract</jats:title><jats:p>We consider the Heisenberg motion groups ℍ𝕄 = ℍ</jats:p>

Unbounded Weyl transform on the Euclidean motion group and Heisenberg motion group

“…In the article [11], it is shown that ρ λ σ are all possible irreducible unitary representations of G that participate in the Plancherel formula. Thus, in view of the above discussion, we shall denote the partial dual of the group G by…”

“…, it follows that f (λ, σ) is a Hilbert-Schmidt operator on H 2 σ . For detailed Fourier analysis on the Heisenberg motion group, see [2,5,11].…”

Uncertainty Principles for Heisenberg Motion Group

“…where dτðλÞ = ð2πÞ −n−1 jλj n dλ is the measure defined on ℝ \ f0g, d σ is the dimension of the space H σ , and k f ðλ, σÞk 2 HS denote the Hilbert-Schmidt norm of f ðλ, σÞ [9]. At the end of this paragraph, we introduce an orthonormal basis for L 2 ðℂ n × KÞ [10].…”

Hardy and Miyachi Theorems for Heisenberg Motion Groups

“…They are enough to describe the Plancherel measure of (cf. [11]). For an integrable function on its Fourier transform is the operator-valued function given by where is the Haar measure on Observe that is a bounded linear operator acting on …”

“…Given , consider the group Fourier transform: where the partial Fourier transform is as in formula (2.2). Then for , one gets as in [11, page 30] that and the Plancherel formula for the group reads: where stands for the dimension of the space of the representation as in the second section. We also introduce the Euclidean norm on as Our first result in this section is the following theorem.…”