Wavelets from Laguerre Polynomials and Toeplitz-Type Operators

Abstract

We study a parameterized family of Toeplitz-type operators with respect to specific wavelets whose Fourier transforms are related to Laguerre polynomials. On the one hand, this choice of wavelets underlines the fact that these operators acting on wavelet subspaces share many properties with the classical Toeplitz operators acting on the Bergman spaces. On the other hand, it enables to study poly-Bergman spaces and Toeplitz operators acting on them from a different perspective. Restricting to symbols depending only on vertical variable in the upper half-plane of the complex plane these operators are unitarily equivalent to a multiplication operator with a certain function. Since this function is responsible for many interesting features of these Toeplitz-type operators and their algebras, we investigate its behavior in more detail. As a by-product we obtain an interesting observation about the asymptotic behavior of true poly-analytic Bergman spaces. Isomorphisms between the Calderón-Toeplitz operator algebras and functional algebras are described and their consequences in time-frequency analysis and applications are discussed.Mathematics Subject Classification (2010). Primary 47B35, 42C40; Secondary 47G30, 47L80.

On Boundedness of Calderón–Toeplitz Operators

“…We present a number of examples and construct wide families of unbounded symbols for which the Calderón-Toeplitz operator is not only bounded, but also belongs to the algebra of bounded Calderón-Toeplitz operators generated by bounded symbols on R + having limits at the endpoints of [0, +∞]. This extends results for bounded symbols from [9] to certain unbounded ones.…”

“…In this paper we continue the detailed study of Calderón-Toeplitz operators T (k) a acting on wavelet subspace A (k) , which was initiated in [9]. For a bounded symbol a on G the Calderón-Toeplitz operator T (k) a is clearly bounded on A (k) .…”

“…This connection was further investigated in [1] where also some interesting applications were given. The specificity of this choice of wavelets ψ (k) , whose Fourier transforms are related to Laguerre polynomials, has enabled us to study in [9] in more detail the family of Calderón-Toeplitz operators T (k) a acting on wavelet subspaces A (k) (with parameter k = 0, 1, 2, . .…”

“…Therefore the aim of this paper is to study in detail the boundedness properties of Calderón-Toeplitz operators with such unbounded symbols and to give sufficient conditions for their simultaneous boundedness on all wavelet subspaces. The main tool in our study is the result, which we have shown in [9], that the Calderón-Toeplitz operator T (k) a acting on A (k) with a symbol a = a(ℑζ), ζ = (u, v), is unitarily equivalent to the multiplication operator γ a,k I acting on…”

Sharp rates of convergence for accumulated spectrograms

“…It is a basic tool in spectral estimation [7], it has been applied to the analysis of brain signals [47] and it is also an important step in the recent high-resolution time-frequency algorithm ConceFT [14]. The accumulated spectrogram has also been investigated in non-Euclidean contexts [20] -see also [25]. Numerical experiments show that when N is close to the critical value ≈ |Ω|, the sum of the N spectrograms that are most concentrated on Ω almost exhaust the domain Ω; i.e., the accumulated spectrogram looks approximately like 1 Ω .…”