Coorbit Theory and Bergman Spaces

Feichtinger, Hans G.; Pap, Margit

doi:10.1007/978-3-319-01806-5_4

Cited by 11 publications

(12 citation statements)

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“…This unified treatment is a fairly novel feature of the theory; prior to [24,25], most sources concerned with coorbit theory for higher-dimensional wavelet transforms concentrated on special cases [7,8,6,9,31,17], each of which was treated with tailor-made approaches. It would be interesting to study further extensions of the method, e.g.…”

Section: Discussionmentioning

confidence: 99%

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Simplified vanishing moment criteria for wavelets over general dilation groups, with applications to abelian and shearlet dilation groups

Führ

Tousi

2017

Applied and Computational Harmonic Analysis

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Abstract. We consider the coorbit theory associated associated to a square-integrable, irreducible quasi-regular representation of a semidirect product group G = R d ⋊ H. The existence of coorbit spaces for this very general setting has been recently established, together with concrete vanishing moment criteria for analyzing vectors and atoms that can be used in the coorbit scheme. These criteria depend on fairly technical assumptions on the dual action of the dilation group, and it is one of the chief purposes of this paper to considerably simplify these assumptions.We then proceed to verify the assumptions for large classes of dilation groups, in particular for all abelian dilation groups in arbitrary dimensions, as well as a class called generalized shearlet dilation groups, containing and extending all known examples of shearlet dilation groups employed in dimensions two and higher. We explain how these groups can be systematically constructed from certain commutative associative algebras of the same dimension, and give a full list, up to conjugacy, of shearing groups in dimensions three and four. In the latter case, three previously unknown groups are found.As a result, the existence of Banach frames consisting of compactly supported wavelets, with simultaneous convergence in a whole range of coorbit spaces, is established for all groups involved.

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Section: Discussionmentioning

confidence: 99%

“…Instead of invoking Lemma 2.8 for the remaining inequalities, we may as well observe directly that being abelian, H is unimodular, and thus (17) holds with e 4 = 0. Furthermore, for h = r · 1 A + a, the fact that det(id + a) = 1 for nilpotent matrices a yields |det(h)| = |r| d , thus (16) holds with e 3 = d.…”

Section: Abelian Dilation Groupsmentioning

confidence: 99%

Simplified vanishing moment criteria for wavelets over general dilation groups, with applications to abelian and shearlet dilation groups

Führ

Tousi

2017

Applied and Computational Harmonic Analysis

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“…The connection with Bergman spaces was to our knowledge initially pointed out in [28,Section 7.3]. To introduce this topic in a brief and succinct manner, we will give an outline of the definitions and results given in [57] and [31]. We encourage the reader to seek out the more recent and technical paper [9] for interesting results in higher dimensions.…”

Section: Bergman Spaces and The Blaschke Groupmentioning

confidence: 99%

A Primer on Coorbit Theory -- From Basics to Recent Developments

Berge

2021

Preprint

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Coorbit theory is a powerful machinery that constructs a family of Banach spaces, the socalled coorbit spaces, from well-behaved unitary representations of locally compact groups. A core feature of coorbit spaces is that they can be discretized in a way that reflects the geometry of the underlying locally compact group. Many established function spaces such as modulation spaces, Besov spaces, Sobolev-Shubin spaces, and shearlet spaces are examples of coorbit spaces.The goal of this survey is to give an overview of coorbit theory with the aim of presenting the main ideas in an accessible manner. Coorbit theory is generally seen as a complicated theory, filled with both technicalities and conceptual difficulties. Faced with this obstacle, we feel obliged to convince the reader of the theory's elegance. As such, this survey is a showcase of coorbit theory and should be treated as a stepping stone to more complete sources. in Section 4.3. We end in Section 4.4 by giving references to recent developments related to embeddings between coorbit spaces and generalizations of coorbit theory.Acknowledgments: I have received advice and concrete suggestions from many researchers throughout the writing process. I would in particular like to thank Stine Marie Berge, Franz Luef, and Felix Voigtlaender for illuminating discussions and helpful comments. Finally, I would like to express my gratitude to everyone who participated in the seminar course I gave on coorbit theory at the Norwegian University of Science and Technology during the fall of 2020. Starting OutWe start by giving an overview of preliminary topics, namely locally compact groups, unitary representations, and basic properties of the (generalized) wavelet transform. Most of this material is fairly standard, and is mainly collected from the books [34,42,16,36,21]. We aim for a suitable generality and present concrete examples as we go along. Prelude on Locally Compact GroupsThe first order of business is to get acquainted with locally compact groups.Definition 2.1. A locally compact group is a locally compact Hausdorff topological space G that is simultaneously a group such that the multiplication and inversion mapsFor us, the object of main importance on a locally compact group is the left Haar measure: Recall that a Radon measure is Borel measure that is finite on compact sets, inner regular on open sets, and outer regular on all Borel sets. Do not worry if you are rusty on the measure-theoretic nonsense; we will never use these technical conditions explicitly. The important point is that each locally compact group G can be equipped with a unique (up to a positive constant) left-invariant Radon measure µ L , that is, µ L satisfies µ L (xE) = µ L (E) for all x ∈ G and every Borel set E ⊂ G. We call the measure µ L the left Haar measure of the group G. The existence of the left Haar measure implies that any locally compact group is canonically equipped with a measure-theoretic setting.As the terminology indicates, there is also a right Haar measure µ R on any locally ...

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“…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]).…”

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

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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.

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Coorbit Theory and Bergman Spaces

Cited by 11 publications

References 33 publications

Simplified vanishing moment criteria for wavelets over general dilation groups, with applications to abelian and shearlet dilation groups

Simplified vanishing moment criteria for wavelets over general dilation groups, with applications to abelian and shearlet dilation groups

A Primer on Coorbit Theory -- From Basics to Recent Developments

Quaternionic Blaschke Group

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