Spectral theorems associated with the directional short-time Fourier transform

In this paper, we are interested in the directional short-time Fourier transform 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 Lp(Rd), 1 <= p <=infinity After wards, 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 multipliers. Moreover we study functions whose time-frequency content are 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.