Two-Wavelet Multipliers on the Dual of the Laguerre Hypergroup and Applications

Abstract

In this paper, we are interested in the Laguerre hypergroup $$\mathbb {K} = [0,\infty )\times {\mathbb {R}}$$ which is the fundamental manifold of the radial function space for the Heisenberg group. So, we consider the generalized shift operator generated by the dual of the Laguerre hypergroup $$\widehat{\mathbb {K}}$$ which can be topologically identified with the so-called Heisenberg fan, the subset of $${\mathbb {R}}^{2}$$ $$\begin{aligned} \cup _{j\in {\mathbb {N}}}\left\{ (\lambda ,\mu )\in {\mathbb {R}}^{2}:\mu =|\lambda |(2j+\alpha +1), \; \lambda \ne 0\right\} \cup \left\{ (0,\mu )\in {\mathbb {R}}^{2}:\mu \ge 0\right\} , \end{aligned}$$ by means of which the notion of a generalized two-wavelet multiplier is investigated. The boundedness and compactness of the generalized two-wavelet multipliers are studied on $$L^{p}_{\alpha }(\mathbb {K})$$ , $$1 \le p \le \infty $$ . Afterwards, we introduce the generalized Landau–Pollak–Slepian operator and we give its trace formula. We show that the generalized two-wavelet multiplier is unitary equivalent to a scalar multiple of the generalized Landau–Pollak–Slepian operator. As applications, we prove an uncertainty principle of Donoho–Stark type involving $$\varepsilon $$ -concentration of the generalized two-wavelet multiplier operators. Moreover, we study functions whose time–frequency content is concentrated in a region with finite measure in phase space using the phase space restriction operators as a main tool. We obtain approximation inequalities for such functions using a finite linear combination of eigenfunctions of these operators.