Reproducing Spaces and Localization Operators

Abstract

This paper, by using of windowed Fourier transform (WFT), gives a family of embedding operators $T_{n} :L^{2} {\left( R \right)} \to L^{2} {\left( {C,e^{{ - \frac{{{\left| z \right|}^{2} }} {2}}} \frac{{dzd\overline{z} }} {{4\pi i}}} \right)}$, s.t. $T_{n} L^{2} {\left( R \right)} \subseteq L^{2} {\left( {C,e^{{ - \frac{{{\left| z \right|}^{2} }} {2}}} \frac{{dzd\overline{z} }} {{4\pi i}}} \right)}$ are reproducing subspaces (n = 0, Bargmann Space); and gives a reproducing kernel and an orthonormal basis (ONB) of T _n L ²(R). Furthermore, it shows the orthogonal spaces decomposition of $L^{2} {\left( {C,e^{{ - \frac{{{\left| z \right|}^{2} }} {2}}} \frac{{dzd\overline{z} }} {{4\pi i}}} \right)}$. Finally, by using the preceding results, it shows the eigenvalues and eigenfunctions of a class of localization operators associated with WFT, which extends the result of Daubechies in [1] and [6].