Karlheinz Gröchenig

Banach spaces related to integrable group representations and their atomic decompositions, I

Abstract We present a general theory of Banach spaces which are invariant under the action of an integrable group representation and give their atomic decompositions with respect to coherent states, i.e., the atoms arise from a single element under the group action. Several well-known decomposition theories are contained as special examples and are unified under the aspect of group theory.

Nonuniform Sampling and Reconstruction in Shift-Invariant Spaces

This article discusses modern techniques for nonuniform sampling and reconstruction of functions in shift-invariant spaces. It is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shift-invariant subspaces by bringing together wavelet theory, frame theory, reproducing kernel Hilbert spaces, approximation theory, amalgam spaces, and sampling. Inspired by applications taken from communication, astronomy, and medicine, the following aspects will be emphasized: (a) The sampling problem is well defined within the setting of shift-invariant spaces. (b) The general theory works in arbitrary dimension and for a broad class of generators. (c) The reconstruction of a function from any sufficiently dense nonuniform sampling set is obtained by efficient iterative algorithms. These algorithms converge geometrically and are robust in the presence of noise. (d) To model the natural decay conditions of real signals and images, the sampling theory is developed in weighted L p-spaces.

Describing Functions: Atomic Decompositions Versus Frames.

The theory of frames and non-orthogonal series expansions with respect to coherent states is extended to a general class of spaces, the so-called coorbit spaces. Special cases include wavelet expansions for the Besov-Triebel-Lizorkin spaces, Gabortype expansions for modulation spaces, and sampling theorems for wavelet and Gabor transforms.

RECONSTRUCTION ALGORITHMS IN IRREGULAR SAMPLING

A constructive solution of the irregular sampling problem for band- limited functions is given. We show how a band-limited function can be com- pletely reconstructed from any random sampling set whose density is higher than the Nyquist rate, and give precise estimates for the speed of convergence of this iteration method. Variations of this algorithm allow for irregular sampling with derivatives, reconstruction of band-limited functions from local averages, and irregular sampling of multivariate band-limited functions. In the irregular sampling problem one is asked whether and how a band- limited function / can be completely reconstructed from its irregularly sam- pled values f(xi). This has many applications in signal and image processing, seismology, meteorology, medical imaging, etc. Finding constructive solutions of this problem has received considerable attention among mathematicians and engineers. The mathematical literature provides several uniqueness results (1, 2, 17, 18, 19). It is now part of the folklore that for stable sampling the sampling rate must be at least the Nyquist rate (18). These results, as deep as they are, have had little impact for the applied sciences, because they were not constructive. If the sampling set is just a perturbation of the regular oversampling, then a reconstruction method has been obtained in a seminal paper by Duffin and Schaeffer (6) (see also (29)): if for some L > 0, a > 0, and o > 0 the sampling points xk , k e Z , satisfy (a) \xk - ok\ a, k ^ I, then the norm equivalence A iR \f(x)\2dx ) with w < n/o. This norm equivalence implies that it is possible to reconstruct / through an iterative procedure, the so-called frame method. Most of the later work on constructive methods consists of variations of this method (3, 21, 22, 26). The above conditions on the sampling set exclude random irregular sampling sets, e.g., sets with regions of higher sampling density. A partial, but undesirable remedy, to handle highly irregular sampling sets, would be to force the above conditions by throwing away information on part of the points and accept a very slow convergence of the iteration.

Wiener's lemma for twisted convolution and Gabor frames

As a consequence, Ca is invertible and bounded on all ?p(Zd) for 1 < p < oo simultaneously. In this article we study several non-commutative generalizations of Wieners Lemma and their application to Gabor theory. The paper is divided into two parts: the first part (Sections 2 and 3) is devoted to abstract harmonic analysis and extends Wieners Lemma to twisted convolution. The second part (Section 4) is devoted to the theory of Gabor frames, specifically to the design of dual windows with good time-frequency localization. In particular, we solve a conjecture of Janssen, Feichtinger and one of us [17], [18], [9]. These two topics appear to be completely disjoint, but they are not. The solution of the conjectures about Gabor frames is an unexpected application of methods from non-commutative harmonic analysis to application-oriented mathematics. It turns out that the connection between twisted convolution and the Heisenberg group and the theory of symmetric group algebras are precisely the tools needed to treat the problem motivated by signal analysis. To be more concrete, we formulate some of our main results first and will deal with the details and the technical background later.

Time–Frequency analysis of localization operators

Abstract We study a class of pseudodifferential operators known as time–frequency localization operators, Anti-Wick operators, Gabor–Toeplitz operators or wave packets. Given a symbol a and two windows ϕ1,ϕ2, we investigate the multilinear mapping from (a,ϕ 1 ,ϕ 2 )∈ S ′( R 2d )× S ( R d )× S ( R d ) to the localization operator Aaϕ1,ϕ2 and we give sufficient and necessary conditions for Aaϕ1,ϕ2 to be bounded or to belong to a Schatten class. Our results are formulated in terms of time–frequency analysis, in particular we use modulation spaces as appropriate classes for symbols and windows.

Iterative reconstruction of multivariate band-limited functions from irregular sampling values

This paper describes a real analysis approach to the problem of complete reconstruction of a band-limited, multivariate function f from irregularly spaced sampling values

Gabor wavelets and the Heisenberg group: Gabor expansions and short time Fourier transform from the group theoretical point of view

(f(x_i ))_{i \in I}

Acceleration of the frame algorithm

. The required sampling density of the set

Time-Frequency Analysis of Sjöstrand's Class

X = (x_i )_{i \in I}