Toeplitz Localization Operators: Spectral Functions Density

We consider two classes of localization operators based on the Calderón and Gabor reproducing formulas and represent them in a uniform way as Toeplitz operators. We restrict our attention to the generating symbols depending on the first coordinate in the phase space. In this case, the Toeplitz localization operators (TLOs) exhibit an explicit diagonalization, i.e., there exists an isometric isomorphism that transforms all TLOs to the multiplication operators by some specific functions—we call them spectral functions. We show that these spectral functions can be written in the form of a convolution of the generating symbol of TLO with a kernel function incorporating an admissible wavelet/window. Using the Wiener’s deconvolution technique on the real line, we prove that the set of spectral functions is dense in the C^∗-algebra of bounded uniformly continuous functions on the real line under the assumption that the Fourier transform of the kernel function does not vanish on the real line. This provides an explicit and independent description of the C^∗-algebra generated by the set of spectral functions. The result is then applied to the case of a parametric family of wavelets related to Laguerre functions. Thereby we also provide an explicit description of the C^∗-algebra generated by vertical Toeplitz operators on true poly-analytic Bergman spaces over the upper half-plane.