Abstract
In this paper, we are interested in the Laguerre hypergroup K = [0, infinity) x 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 (K) over cap which can be topologically identified with the so-called Heisenberg fan, the subset of R-2.
boolean OR(j is an element of N) {(lambda, mu) is an element of R-2 : mu = vertical bar lambda vertical bar(2j + alpha + 1), lambda not equal 0} boolean OR {(0, mu) is an element of R-2 : mu >= 0, 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-alpha(p)(K), 1 <= p <= infinity. 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 epsilon-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.