On Toeplitz-type Operators Related to Wavelets

Abstract

Let G be the "ax + b"-group with the left invariant Haar measure dν and ψ be a fixed real-valued admissible wavelet on L2(R). The structure of the space of Calderón (wavelet) transforms W ψ (L2(R)) inside L2(G, dν) is described. Using this result some representations, properties and the Wick calculus of the Calderón-Toeplitz operators Ta acting on W ψ (L2(R)) whose symbols a = a(ζ) depend on v = ζ for ζ ∈ G are investigated. (2000). Primary 46E22, 47B35; Secondary 42C40. Mathematics Subject Classification

Erlangen Program at Large: An Overview

“…Thus the complex analysis become decoupled from the traditional wavelet theory. As a result the application of wavelet theory shall relay on an extraneous mother wavelets [47].…”

On the structure of the space of wavelet transforms

“…Note that the above representation of the space of Calderón transforms W ψ (L 2 (R)) is especially important in the study of the Calderón-Toeplitz operators with symbols depending only on imaginary part of ζ = (u, v) ∈ G, cf. [3]. Related operators show up in physics when working with coherent states, cf.…”

On Toeplitz localization operators

“…. In what follows we show that the Bargmann-type transform R Ψ essentially simplifies the previous computations made in [4] and [5], and enables to obtain many interesting results for the Toeplitz localization operators which both cases share in common in a more transparent way.…”

“…given by dy) we have obtained the structural result saying "how much space occupies the subspace W Ψ (L 2 (R)) inside L 2 (G, dζ)", see [4,Theorem 3.3] and [5, Theorem 1] for more details. Now the trick is that in comparison with our previous approach the second operator V Ψ in both cases is not needed to study the TLO's T Ψ a , thus providing a much easier way to the properties of T Ψ a .…”