Properties of the voice transform of the Blaschke group and connections with atomic decomposition results in the weighted Bergman spaces

Abstract. This article describes how the ideas promoted by the fundamental papers published by M. Frazier and B. Jawerth in the eighties have influenced subsequent developments related to the theory of atomic decompositions and Banach frames for function spaces such as the modulation spaces and BesovTriebel-Lizorkin spaces.Both of these classes of spaces arise as special cases of two different, general constructions of function spaces: coorbit spaces and decomposition spaces. Coorbit spaces are defined by imposing certain decay conditions on the so-called voice transform of the function/distribution under consideration. As a concrete example, one might think of the wavelet transform, leading to the theory of Besov-Triebel-Lizorkin spaces.Decomposition spaces, on the other hand, are defined using certain decompositions in the Fourier domain. For Besov-Triebel-Lizorkin spaces, one uses a dyadic decomposition, while a uniform decomposition yields modulation spaces.Only recently, the second author has established a fruitful connection between modern variants of wavelet theory with respect to general dilation groups (which can be treated in the context of coorbit theory) and a particular family of decomposition spaces. In this way, optimal inclusion results and invariance properties for a variety of smoothness spaces can be established. We will present an outline of these connections and comment on the basic results arising in this context.

show abstract

“…It was also used later on in the context of the Blaschke group by Margit Pap and Ferenc Schipp (see e.g. [61,59,60]).…”

Section: Thementioning

confidence: 99%

“…Yet another group, the so-called Blaschke group, is in the background of a series of papers by M. Pap and her coauthors ( [61,59,60,34]).…”

Section: Shearlets and Other Constructionsmentioning

confidence: 99%

From Frazier-Jawerth characterizations of Besov spaces to wavelets and decomposition spaces

Feichtinger

Voigtlaender²

2017

Functional Analysis, Harmonic Analysis, And Image Processing

View full text Add to dashboard Cite

show abstract

“…We also obtain frame expansions for the Bergman spaces, which is related to the existence of sampling sequences. We are not aware of any results in this direction except in the special case of the unit disc [23,38,39], though they are hardly surprising considering the work by [42,16].…”

Section: Introductionmentioning

confidence: 99%

New atomic decompositions for Bergman spaces on the unit ball

Christensen¹,

Gröchenig²,

Ólafsson³

2017

Indiana Univ. Math. J.

View full text Add to dashboard Cite

We derive atomic decompositions and frames for weighted Bergman spaces of several complex variables on the unit ball in the spirit of Coifman, Rochberg, and Luecking. In contrast to our predecessors, we use group theoretic methods, in particular the representation theory of the discrete series of SU(n, 1) and its covering groups. One of the benefits is a much larger class of admissible atoms.2010 Mathematics Subject Classification. Primary 32A36, 43A15, 42B35, 46E15; Secondary 22D12.

show abstract

“…Using the parametrization of the Blaschke group reflects better in the same time the properties of the covering group and the action of the representations on different analytic function spaces, see [6], where it is explained in detail the relation between SU(1, 1) and the Blaschke group, and why we consider the Blaschke group useful in order to develop wavelet analysis on this group. One reason is that the techniques of the complex analysis can be applied more directly in the study of the properties of the voice transforms (so called hyperbolic wavelet transforms) generated by representations of the Blaschke group on different analytic function spaces (see [7][8][9][10][11]). The discretization of these special wavelet transforms leads to the construction of analytic rational orthogonal wavelets, and multiresolution analysis (MRA) in the Hardy space of the unit disc, upper half plane, and in weighted Bergman spaces (see [11][12][13][14]).…”

Section: Introductionmentioning

confidence: 99%

“…An important consequence of this relation is the addition formula for these functions (see [7,8,11]). In the same time using the parametrization of the Blaschke group it was easier to apply the coorbit theory (see [15]) in order to obtain atomic decompositions in weighted Bergman spaces (see [6,10]). In this way as a special case we get back well known atomic decompositions in the weighted Bergman spaces obtained by complex techniques, but in addition some new atomic decompositions can be presented.…”

Section: Introductionmentioning

confidence: 99%

Quaternionic Blaschke Group

Pap

Schipp

2018

Mathematics

Self Cite

View full text Add to dashboard Cite

In the complex case, the Blaschke group was introduced and studied. It turned out that in the complex case this group plays important role in the construction of analytic wavelets and multiresolution analysis in different analytic function spaces. The extension of the wavelet theory to quaternion variable function spaces would be very beneficial in the solution of many problems in physics. A first step in this direction is to give the quaternionic analogue of the Blaschke group. In this paper we introduce the quaternionic Blaschke group and we study the properties of this group and its subgroups.

show abstract

Properties of the voice transform of the Blaschke group and connections with atomic decomposition results in the weighted Bergman spaces

Cited by 17 publications

References 6 publications

From Frazier-Jawerth characterizations of Besov spaces to wavelets and decomposition spaces

From Frazier-Jawerth characterizations of Besov spaces to wavelets and decomposition spaces

New atomic decompositions for Bergman spaces on the unit ball

Quaternionic Blaschke Group

Contact Info

Product

Resources

About