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Let (\gnm)n,m ? \Zst(\gnm)_{n,m\in\Zst} be a Gabor frame for \LtR\LtR for given window gg. We show that the window \ho = \SQI g\ho=\SQI g that generates the canonically associated tight Gabor frame minimizes ||g-h||\|g-h\| among all windows hh generating a normalized tight Gabor frame. We present and prove versions of this result in the time domain, the frequency domain, the time-frequency domain, and the Zak transform domain, where in each domain the canonical \ho\ho is expressed using functional calculus for Gabor frame operators. Furthermore, we derive a Wiener--Levy type theorem for rationally oversampled Gabor frames. Finally, a Newton-type method for a fast numerical calculation of \ho\ho is presented. We analyze the convergence behavior of this method and demonstrate the efficiency of the proposed algorithm by some numerical examples. 相似文献
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We settle a conjecture of Joseph Renes about the existence and construction of certain equiangular tight frames. 相似文献
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Ohne Zusammenfassung 相似文献
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J. Formanék J. E. Politis R. Geduldt Sidersky E. Wein A. Herzfeld Burkhard Bodenbender Scheller E. Parcus Striegler Pellet Delville Preuss Gerken Bruhns Volpert F. G. Wiechmann Dammüller Gantenberg R. Creydt Wohl Strohmer Cech Herles Lindet Courtonne Wolf 《Analytical and bioanalytical chemistry》1891,30(1):63-89
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Journal of Fourier Analysis and Applications - Assume we are given a sum of linear measurements of s different rank-r matrices of the form $$\varvec{y}= \sum _{k=1}^{s} \mathcal {A}_k... 相似文献
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Karlheinz Gröchenig Ziemowit Rzeszotnik Thomas Strohmer 《Integral Equations and Operator Theory》2010,67(2):183-202
The finite section method is a classical scheme to approximate the solution of an infinite system of linear equations. Based
on an axiomatic framework we present a convergence analysis of the finite section method for unstructured matrices on weighted
ℓ
p
-spaces. In particular, the stability of the finite section method on ℓ
2 implies its stability on weighted ℓ
p
-spaces. Our approach uses recent results from the theory of Banach algebras of matrices with off-diagonal decay. Furthermore,
we demonstrate that Banach algebra theory provides a natural framework for deriving a finite section method that is applicable
to large classes of unstructured non-hermitian matrices as well as to least squares problems. 相似文献
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Max Hügel Holger Rauhut Thomas Strohmer 《Foundations of Computational Mathematics》2014,14(1):115-150
We consider the problem of detecting the locations of targets in the far field by sending probing signals from an antenna array and recording the reflected echoes. Drawing on key concepts from the area of compressive sensing, we use an ? 1-based regularization approach to solve this, generally ill-posed, inverse scattering problem. As is common in compressive sensing, we exploit randomness, which in this context comes from choosing the antenna locations at random. With n antennas we obtain n 2 measurements of a vector $x \in\mathbb{C}^{N}$ representing the target locations and reflectivities on a discretized grid. It is common to assume that the scene x is sparse due to a limited number of targets. Under a natural condition on the mesh size of the grid, we show that an s-sparse scene can be recovered via ? 1-minimization with high probability if n 2≥Cslog2(N). The reconstruction is stable under noise and when passing from sparse to approximately sparse vectors. Our theoretical findings are confirmed by numerical simulations. 相似文献
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F. Strohmer 《Monatshefte für Chemie / Chemical Monthly》1881,2(1):539-540
Ohne Zusammenfassung 相似文献