The Experts below are selected from a list of 12 Experts worldwide ranked by ideXlab platform
Alexander Lindner - One of the best experts on this subject based on the ideXlab platform.
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Lower Bounds for finite wavelet and Gabor systems
Wavelet Applications in Signal and Image Processing VIII, 2000Co-Authors: Ole Christensen, Alexander LindnerAbstract:For a function ΦeL 2 ( R ) and parameters a >1, b >0, the corresponding wavelet family is the set of functions (function in paper). We show that for a dense set of functions (function in paper), every finite subfamily of the functions (function in paper) will be linearly independent. Under certain conditions, the result is also true for finite subsets (function in paper) of the irregular wavelet system (formula in paper), where (formula in paper). We estimate the corresponding lower Riesz Bound, i.e., we find a positive number A such that (formula in paper) for all finite sequences (function in paper). We discuss cases from wavelet theory where it is crucial to have such estimates. We consider the same question for a Gabor family (formula in paper). A conjecture by Heil, Ramanathan and Topiwala states that when (formula in paper) and (formula in paper) consists of distinct points, then (formula in paper) will automatically be linearly independent. It is known that the conjecture holds in some special cases, e.g., under the assumption that g has support in a half line. We estimate the lower Riesz Bound in that case.
Ole Christensen - One of the best experts on this subject based on the ideXlab platform.
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Lower Bounds for finite wavelet and Gabor systems
Wavelet Applications in Signal and Image Processing VIII, 2000Co-Authors: Ole Christensen, Alexander LindnerAbstract:For a function ΦeL 2 ( R ) and parameters a >1, b >0, the corresponding wavelet family is the set of functions (function in paper). We show that for a dense set of functions (function in paper), every finite subfamily of the functions (function in paper) will be linearly independent. Under certain conditions, the result is also true for finite subsets (function in paper) of the irregular wavelet system (formula in paper), where (formula in paper). We estimate the corresponding lower Riesz Bound, i.e., we find a positive number A such that (formula in paper) for all finite sequences (function in paper). We discuss cases from wavelet theory where it is crucial to have such estimates. We consider the same question for a Gabor family (formula in paper). A conjecture by Heil, Ramanathan and Topiwala states that when (formula in paper) and (formula in paper) consists of distinct points, then (formula in paper) will automatically be linearly independent. It is known that the conjecture holds in some special cases, e.g., under the assumption that g has support in a half line. We estimate the lower Riesz Bound in that case.
Noli N. Reyes - One of the best experts on this subject based on the ideXlab platform.
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ZEROS OF FINITE WAVELET SUMS
Taiwanese Journal of Mathematics, 2005Co-Authors: Noli N. ReyesAbstract:For certain analytic functions $ \psi $, a lower Riesz Bound for a finite wavelet system generated by $ \psi $, yields an upper Bound for the number of zeros on a Bounded interval of the corresponding wavelet sums. In particular, we show that if the Fourier transform of $ \psi $ is compactly supported, say on $ [ -\Omega , \Omega ] ,$ and if $B > 2 e \Omega $, then any finite sum $ \sum_{ |k | \leq \alpha / 2 } a_{k} \psi (x- k) $ cannot have more than $ B \alpha $ zeros in $ [ -\alpha , \alpha ] $ for $ \alpha >0 $ sufficiently large.
Karlheinz Gröchenig - One of the best experts on this subject based on the ideXlab platform.
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Linear independence of time-frequency shifts?
Monatshefte für Mathematik, 2015Co-Authors: Karlheinz GröchenigAbstract:We investigate finite sections of Gabor frames and study the asymptotic behavior of their lower Riesz Bound. From a numerical point of view, these sets of time-frequency shifts are linearly dependent, whereas from a rigorous analytic point of view, they are conjectured to be linearly independent.
Shaofan Li - One of the best experts on this subject based on the ideXlab platform.
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Moving least-square reproducing kernel method Part II: Fourier analysis
Computer Methods in Applied Mechanics and Engineering, 1996Co-Authors: Shaofan LiAbstract:In Part I of this work, the moving least-square reproducing kernel (MLSRK) method is formulated and implemented. Based on its generic construction, an m-consistency structure is discovered and the convergence theorems are established. In this part of the work, a systematic Fourier analysis is employed to evaluate and further establish the method. The preliminary Fourier analysis reveals that the MLSRK method is stable for sufficiently dense, non-degenerated particle distribution, in the sense that the kernel function family satisfies the Riesz Bound. One of the novelties of the current approach is to treat the MLSRK method as a variant of the ‘standard’ finite element method and depart from there to make a connection with the multiresolution approximation. In the spirits of multiresolution analysis, we propose the following MLSRK transformation, Fm,kϱ,hu=∑i=1npu,K,ϱiKϱh(x−xi,x)w1 The highlight of this paper is to embrace the MLSRK formulation with the notion of the controlled fp-approximation. Based on its characterization, the Strang-Fix condition for example, a systematic procedure is proposed to design new window functions so they can enhance the computational performance of the MLSRK algorithm. The main effort here is to obtain a constant correction function in the interior region of a general domain, i.e. Cρh = 1. This can create a leap in the approximation order of the MLSRK algorithm significantly, if a highly smooth window function is embedded within the kernel. One consequence of this development is the synchronized convergence phenomenon—a unique convergence mechanism for the MLSRK method, i.e. by properly tuning the dilation parameter, the convergence rate of higher-order error norms will approach the same order convergence rate of the L2 error norm—they are synchronized.
