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1.
Karlheinz Gröchenig 《Monatshefte für Mathematik》1985,100(3):171-182
We study the properties of sequences (c
n
) in a compact groupG such that (x
n
) being (weakly) well-distributed implies (c
n
x
n
) being (weakly) well-distributed and give a complete characterisation in terms of almost constant sequences as well as a generalisation to non-compact locally compact groups. 相似文献
2.
Karlheinz Gröchenig 《Journal of Fourier Analysis and Applications》2014,20(4):865-895
This is a survey about the theory of Gabor frames. We review the structural results about Gabor frames over a lattice and then discuss the few known results about the fine structure of Gabor frames. We add a new result about the relation between properties of the window and properties of the frame set and conclude with a vision of how a more complete theory of the fine structure might look like. 相似文献
3.
Irregular sampling, Toeplitz matrices, and the approximation of entire functions of exponential type
In many applications one seeks to recover an entire function of exponential type from its non-uniformly spaced samples. Whereas the mathematical theory usually addresses the question of when such a function in can be recovered, numerical methods operate with a finite-dimensional model. The numerical reconstruction or approximation of the original function amounts to the solution of a large linear system. We show that the solutions of a particularly efficient discrete model in which the data are fit by trigonometric polynomials converge to the solution of the original infinite-dimensional reconstruction problem. This legitimatizes the numerical computations and explains why the algorithms employed produce reasonable results. The main mathematical result is a new type of approximation theorem for entire functions of exponential type from a finite number of values. From another point of view our approach provides a new method for proving sampling theorems.
4.
We propose a new notion of variable bandwidth that is based on the spectral subspaces of an elliptic operator where p > 0 is a strictly positive function. Denote by the orthogonal projection of Ap corresponding to the spectrum of Ap in ; the range of this projection is the space of functions of variable bandwidth with spectral set in Λ. We will develop the basic theory of these function spaces. First, we derive (nonuniform) sampling theorems; second, we prove necessary density conditions in the style of Landau. Roughly, for a spectrum the main results say that, in a neighborhood of , a function of variable bandwidth behaves like a band‐limited function with local bandwidth . Although the formulation of the results is deceptively similar to the corresponding results for classical band‐limited functions, the methods of proof are much more involved. On the one hand, we use the oscillation method from sampling theory and frame‐theoretic methods; on the other hand, we need the precise spectral theory of Sturm‐Liouville operators and the scattering theory of one‐dimensional Schrödinger operators. © 2017 Wiley Periodicals, Inc. 相似文献
5.
Karlheinz Gröchenig 《Advances in Computational Mathematics》2003,18(2-4):149-157
It is shown that every localized frame is a finite union of Riesz sequences. This is a partial solution to a question of Feichtinger. 相似文献
6.
Elena Cordero Karlheinz Grö chenig 《Proceedings of the American Mathematical Society》2005,133(12):3573-3579
We study time-frequency localization operators of the form , where is the symbol of the operator and are the analysis and synthesis windows, respectively. It is shown in an earlier paper by the authors that a sufficient condition for , the Schatten class of order , is that belongs to the modulation space and the window functions to the modulation space . Here we prove a partial converse: if for every pair of window functions with a uniform norm estimate, then the corresponding symbol must belong to the modulation space . In this sense, modulation spaces are optimal for the study of localization operators. The main ingredients in our proofs are frame theory and Gabor frames. For and , we recapture earlier results, which were obtained by different methods.
7.
We introduce a new notion for the deformation of Gabor systems. Such deformations are in general nonlinear and, in particular, include the standard jitter error and linear deformations of phase space. With this new notion we prove a strong deformation result for Gabor frames and Gabor Riesz sequences that covers the known perturbation and deformation results. Our proof of the deformation theorem requires a new characterization of Gabor frames and Gabor Riesz sequences. It is in the style of Beurling's characterization of sets of sampling for bandlimited functions and extends significantly the known characterization of Gabor frames “without inequalities” from lattices to non-uniform sets. 相似文献
8.
It is shown that local Fourier bases are unconditional bases for the modulation spaces on R, including the Bessel potential spaces and the Segal algebra S
0
. As a consequence, the abstract function spaces, that are defined by the approximation properties with respect to a local
Fourier basis, are precisely the modulation space s.
April 22, 1998. Date accepted: May 18, 1999. 相似文献
9.
10.
Karlheinz Grö chenig Michael Leinert 《Journal of the American Mathematical Society》2004,17(1):1-18
We prove non-commutative versions of Wiener's Lemma on absolutely convergent Fourier series (a) for the case of twisted convolution and (b) for rotation algebras. As an application we solve some open problems about Gabor frames, among them the problem of Feichtinger and Janssen that is known in the literature as the ``irrational case'.