Hans G. Feichtinger

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.

Frame-theoretic analysis of oversampled filter banks

We provide a frame-theoretic analysis of oversampled finite impulse response (FIR) and infinite impulse response (FIR) uniform filter banks (FBs). Our analysis is based on a new relationship between the FBs polyphase matrices and the frame operator corresponding to an FB. For a given oversampled analysis FB, we present a parameterization of all synthesis FBs providing perfect reconstruction. We find necessary and sufficient conditions for an oversampled FB to provide a frame expansion. A new frame-theoretic procedure for the design of paraunitary FBs from given nonparaunitary FBs is formulated. We show that the frame bounds of an FB can be obtained by an eigen-analysis of the polyphase matrices. The relevance of the frame bounds as a characterization of important numerical properties of an FB is assessed by means of a stochastic sensitivity analysis. We consider special cases in which the calculation of the frame bounds and synthesis filters is simplified. Finally, simulation results are presented.

Advances in Gabor Analysis

From the Publisher: Gabor Analysis constitutes a central part of time-frequency analysis. While preserving the symmetry between the time (location) domain and the frequency (wave number) domain it avoids the high degree of redundancy inherent in the continuous short-time Fourier transform. The ability to resolve details of a signal ( or a system impulse response) in a two-dimensional representation, whose coefficients have a very natural interpretation, is the basis for many interesting applications in electrical engineering, or signal-and image processing in general, and thus makes Gabor Analysis an important branch of applied mathematics, whose full recognition lies ahead of us. The proposed book contributes positively to this devleopment. The aim of the book is to provide an overview of recent developments in the area of Gabor analysis by bringing together the leading scientists of various disciplines related to this subject. Covering theory, numerics, as well as applications of Gabor analysis, the book will not merely be a collection of articles but a nice blend of invited chapters with a consistent presentation. Each chapter contains the most recent research while providing enough background material to put the work into proper context. Graduate students, practitioners and other researchers in the areas of numerical analysis, electrical engineering and applied mathematics will find this book a current and useful resource.

Banach spaces related to integrable group representations and their atomic decompositions. Part II

We continue the investigation of coorbit spaces which can be attached to every integrable, irreducible, unitary representation of a locally compact groupG and every reasonable function space onG. Whereas Part I was devoted to atomic decompositions of such spaces, Part II deals with general properties of these spaces as Banach spaces. Among other things we show that inclusions, the quality of embeddings, reflexivity and minimality and maximality of coorbit spaces can be completely characterized by the same properties of the corresponding sequence spaces. In concrete examples (cf. Part III) one recovers several and often difficult theorems with ease.

A Banach space of test functions for Gabor analysis

We introduce the Banach space S 0 ⊑ L 2 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 careful analysis of both the coefficient and the synthesis mapping in Gabor theory shows that an arbitrary window in S 0 not only is a Bessel atom with respect to arbitrary time-frequency lattices, but also yields boundedness between S 0 and e 1. On the other hand, we can study properties of general L 2-atoms since they induce mappings from S 0 to S′0. This enables us to introduce a new, very natural concept of weak duality of Gabor atoms, applying also to the classical pair of the Gauss-function and its dual function determined by Bastiaans. Using the established results, we show a variety of properties that are desirable in applications, like the continuous dependence of the canonical dual window on the given Gabor window and on the lattice; continuity of thresholding and masking operators from signal processing; and an algorithm for the reconstruction of bandlimited functions from samples of the Gabor transform in a corresponding horizontal strip in the time-frequency plane. We also present an approximate Balian-Low Theorem stating that for close-to-critical lattices, the dual Gabor atoms progressively lose their time-frequency localization.

Analysis, Synthesis, and Estimation of Fractal-Rate Stochastic Point Processes

Fractal and fractal-rate stochastic point processes (FSPPs and FRSPPs) provide useful models for describing a broad range of diverse phenomena, including electron transport in amorphous semiconductors, computer-network traffic, and sequences of neuronal action potentials. A particularly useful statistic of these processes is the fractal exponent α, which may be estimated for any FSPP or FRSPP by using a variety of statistical methods. Simulated FSPPs and FRSPPs consistently exhibit bias in this fractal exponent, however, rendering the study and analysis of these processes non-trivial. In this paper, we examine the synthesis and estimation of FRSPPs by carrying out a systematic series of simulations for several different types of FRSPP over a range of design values for α. The discrepancy between the desired and achieved values of α is shown to arise from finite data size and from the character of the point-process generation mechanism. In the context of point-process simulation, reduction of this discrepan...

Exact iterative reconstruction algorithm for multivariate irregularly sampled functions in spline-like spaces: The

We prove that the exact reconstruction of a function s from its samples s(xi) on any “sufficiently dense” sampling set {xi}i∈Λ can be obtained, as long as s is known to belong to a large class of spline-like spaces in Lp(Rn). Moreover, the reconstruction can be implemented using fast algorithms. Since a limiting case is the space of bandlimited functions, our result generalizes the classical Shannon-Whittaker sampling theorem on regular sampling and the Paley-Wiener theorem on non-uniform sampling.

L^p

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

-theory

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

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

. The required sampling density of the set