Spectral theorems associated with the (k, a)-generalized wavelet multipliers

Abstract

We introduce the notion of the (k, a)-generalized wavelet multipliers. Particular cases of such generalized wavelet multipliers are the classical and Dunkl wavelet multipliers. The restriction of the (k, a)-generalized wavelet multipliers to radial functions is given by the generalized Hankel wavelet multiplier. We study the boundedness, Schatten class properties of the (k, a)-generalized wavelet multipliers and we give them trace formula. We prove that the generalized Landau–Pollak–Slepian operator is a (k, a)-generalized wavelet multiplier. Next, we give results on the boundedness and compactness of (k, a)-generalized wavelet multipliers on \(L^{p}_{k,a}(\mathbb {R}^{d})\), \(1 \le p \le \infty \).