The $$\alpha $$ α -modulation transform: admissibility, coorbit theory and frames of compactly supported functions

Abstract

The α-modulation transform is a time-frequency transform generated by square-integrable representations of the affine Weyl-Heisenberg group modulo suitable subgroups. In this paper we prove new conditions that guarantee the admissibility of a given window function. We also show that the generalized coorbit theory can be applied to this setting, assuming specific regularity of the windows. This then yields canonical constructions of Banach frames and atomic decompositions in α-modulation spaces. In particular, we prove the existence of compactly supported (in time domain) vectors that are admissible and satisfy all conditions within the coorbit machinery, which considerably go beyond known results.Math Subject Classification: 42C15, 42C40, 46E15 (primary), 46E35, 57S25 (secondary).

A Guide to Localized Frames and Applications to Galerkin-Like Representations of Operators

“…in [75,60]. Currently the potential of other frames is investigated for solving opera-tor equations in acoustics, such as α-modulation frames [74]. Note, however, that neither wavelet frames nor α-modulation frames are localized in the sense of Definition 1.…”

Structured, compactly supported Banach frame decompositions of decomposition spaces

“…In addition to the constructions by Borup, Nielsen and Rasmussen, Banach frames for α-modulation spaces have also been considered by Fornasier [30], by Dahlke et al [11] and finally by Speckbacher et al [78], all for the case d = 1 and p, q ∈ [1, ∞]. The idea in [30] is to show that a family Γ (δ) α similar to Γ (δ) from Theorem 7.7 is intrinsically self-localized, under suitable readily verifiable assumptions on the generator γ, so that, for a sufficiently small sampling density, the family Γ (δ) α forms a Banach frame and an atomic decomposition for the α-modulation space M p,q s,α (R).…”

Design and properties of wave packet smoothness spaces

“…Decomposition spaces, originally introduced by Feichtinger and Gröbner [16], provide a unified generalisation of modulation and Besov spaces and were used to introduce the α-modulation spaces [25], which have recently received great attention [36,55,32,3,19,31,33,13].…”

“…2) In Subsection 6.5, we shall see that the wave packet smoothness spaces W p,q s (α, α) are identical to the α-modulation spaces M s,α p,q (R 2 ) introduced in Gröbner's PhD thesis [25] and studied further in [3,19,55,33,36,32,61]. Therefore, Theorem 6.5 can be seen as a generalisation of the characterisation of the embeddings between α-modulation spaces, which were first studied in [25,33] and fully understood in [60,32,61].…”