From Frazier-Jawerth characterizations of Besov spaces to Wavelets and Decomposition spaces

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 Besov-Triebel-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

A method for optimizing the ambiguity function concentration

International audienceIn the context of signal analysis and transformation in the time-frequency (TF) domain, controlling the shape of a waveform in this domain is an important issue. Depending on the application, a notion of optimal function may be defined through the properties of the ambiguity function. We present an iterative method for providing such optimal functions under a general concentration constraint of the ambiguity function. At each iteration, it follows a variational approach which maximizes the ambiguity localization via a user-defined weight function F . Under certain assumptions on this latter function, it converges to a waveform which is optimal according to the localization criterion defined by F

Homogeneous Banach Spaces as Banach Convolution Modules over <b><i>M</i></b>(<i>G</i>)

This paper is supposed to form a keystone towards a new and alternative approach to Fourier analysis over LCA (locally compact Abelian) groups G. In an earlier paper the author has already shown that one can introduce convolution and the Fourier–Stieltjes transform on (M(G),∥·∥M), the space of bounded measures (viewed as a space of linear functionals) in an elementary fashion over Rd. Bounded uniform partitions of unity (BUPUs) are easily constructed in the Euclidean setting (by dilation). Moving on to general LCA groups, it becomes an interesting challenge to find ways to construct arbitrary fine BUPUs, ideally without the use of structure theory, the existence of a Haar measure and even Lebesgue integration. This article provides such a construction and demonstrates how it can be used in order to show that any so-called homogeneous Banach space(B,∥·∥B) on G, such as (Lp(G),∥·∥p), for 1≤p∞, or the Fourier–Stieltjes algebra FM(G), and in particular any Segal algebra is a Banach convolution module over (M(G),∥·∥M) in a natural way. Via the Haar measure we can then identify L1(G),∥·∥1 with the closure (of the embedded version) of Cc(G), the space of continuous functions with compact support, in (M(G),∥·∥M), and show that these homogeneous Banach spaces are essentialL1(G)-modules. Thus, in particular, the approximate units act properly as one might expect and converge strongly to the identity operator. The approach is in the spirit of Hans Reiter, avoiding the use of structure theory for LCA groups and the usual techniques of vector-valued integration via duality. The ultimate (still distant) goal of this approach is to provide a new and elementary approach towards the (extended) Fourier transform in the setting of the so-called Banach–Gelfand triple(S0,L2,S0′)(G), based on the Segal algebra S0(G). This direction will be pursued in subsequent papers

A Banach space of test functions for Gabor analysis

We introduce the Banach space S 0 # L which has a variety of properties making it a useful tool in Gabor analysis. S 0 can be characterized as the smallest time-frequency homogeneous Banach space of (continuous) functions. We also present other characterizations of S 0 turning it into a very flexible tool for Gabor analysis and allowing for simplifications of various proofs. A carefu

From Frazier-Jawerth characterizations of Besov spaces to Wavelets and Decomposition spaces

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 Besov-Triebel-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

An Exotic Minimal Banach Space of Functions

This note describes a new Banach space B_0 of square integrable functions on R having many interesting invariance properties. In fact, the Fourier transform, time-frequency shifts, L²-normalized dilations act isometrically on it. For its definition, we make use of a general construction principle for minimal invariant spaces. We demonstrate a variety of properties following immediately from this principle. Furthermore, we give a number of di#erent characterizations, including various atomic decompositions, as well as natural necessary and su#cient conditions for an L²-function to belong to this new space. It turns out that this new space is somewhat exotic, since it is neither rearrangement invariant nor solid

GROUP SPARSITY METHODS FOR COMPRESSIVE CHANNEL ESTIMATION IN DOUBLY DISPERSIVE MULTICARRIER SYSTEMS

We propose advanced compressive estimators of doubly dispersive channels within multicarrier communication systems (including classical OFDM systems). The performance of compressive channel estimation has been shown to be limited by leakage components impairing the channel’s effective delay-Doppler sparsity. We demonstrate a group sparse structure of these leakage components and apply recently proposed recovery techniques for group sparse signals. We also present a basis optimization method for enhancing group sparsity. Statistical knowledge about the channel can be incorporated in the basis optimization if available. The proposed estimators outperform existing compressive estimators with respect to estimation accuracy and, in one instance, also computational complexity. Index Terms — OFDM, multicarrier modulation, channel estimation, doubly dispersive channel, doubly selective channel, compresse

Advances in Computational Mathematics, Special Issue on Localization, Diversity and Uncertainty in Signal Representations

International audienc

Билатерален научноистражувачки проект со Австрија „Асимптотики во коорбит простори”

Во овој проект ќе се поврзат две важни области: хармониска анализа и теорија на дистрибуции, кои имаат непосредна примена во многу други научни области како што се обработка на сигнали, геофизика, анализа на слики и теорија на псеудо-диференцијални оператори. Главна цел на проектот е спроведување асимптотска анализа во коорбит простори. Овие простори се дефинирани преку voice трансформацијата, чии важни примери се вејвлет трансформацијата и short-time Фуриеовата трансформација (STFT) кои се многу применливи алатки во горе споменатите области