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Search: MSC category 42C15 ( General harmonic expansions, frames )

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1. CMB 2016 (vol 59 pp. 528)

Jahan, Qaiser
 Characterization of Low-pass Filters on Local Fields of Positive Characteristic In this article, we give necessary and sufficient conditions on a function to be a low-pass filter on a local field $K$ of positive characteristic associated to the scaling function for multiresolution analysis of $L^2(K)$. We use probability and martingale methods to provide such a characterization. Keywords:multiresolution analysis, local field, low-pass filter, scaling function, probability, conditional probability and martingalesCategories:42C40, 42C15, 43A70, 11S85

2. CMB 2013 (vol 57 pp. 463)

Bownik, Marcin; Jasper, John
 Constructive Proof of Carpenter's Theorem We give a constructive proof of Carpenter's Theorem due to Kadison. Unlike the original proof our approach also yields the real case of this theorem. Keywords:diagonals of projections, the Schur-Horn theorem, the Pythagorean theorem, the Carpenter theorem, spectral theoryCategories:42C15, 47B15, 46C05

3. CMB 2013 (vol 57 pp. 254)

Christensen, Ole; Kim, Hong Oh; Kim, Rae Young
 On Parseval Wavelet Frames with Two or Three Generators via the Unitary Extension Principle The unitary extension principle (UEP) by Ron and Shen yields a sufficient condition for the construction of Parseval wavelet frames with multiple generators. In this paper we characterize the UEP-type wavelet systems that can be extended to a Parseval wavelet frame by adding just one UEP-type wavelet system. We derive a condition that is necessary for the extension of a UEP-type wavelet system to any Parseval wavelet frame with any number of generators, and prove that this condition is also sufficient to ensure that an extension with just two generators is possible. Keywords:Bessel sequences, frames, extension of wavelet Bessel system to tight frame, wavelet systems, unitary extension principleCategories:42C15, 42C40

4. CMB 2013 (vol 56 pp. 729)

Currey, B.; Mayeli, A.
 The Orthonormal Dilation Property for Abstract Parseval Wavelet Frames In this work we introduce a class of discrete groups containing subgroups of abstract translations and dilations, respectively. A variety of wavelet systems can appear as $\pi(\Gamma)\psi$, where $\pi$ is a unitary representation of a wavelet group and $\Gamma$ is the abstract pseudo-lattice $\Gamma$. We prove a condition in order that a Parseval frame $\pi(\Gamma)\psi$ can be dilated to an orthonormal basis of the form $\tau(\Gamma)\Psi$ where $\tau$ is a super-representation of $\pi$. For a subclass of groups that includes the case where the translation subgroup is Heisenberg, we show that this condition always holds, and we cite familiar examples as applications. Keywords:frame, dilation, wavelet, Baumslag-Solitar group, shearletCategories:43A65, 42C40, 42C15

5. CMB 2008 (vol 51 pp. 348)

Casazza, Peter G.; Christensen, Ole
 The Reconstruction Property in Banach Spaces and a Perturbation Theorem Perturbation theory is a fundamental tool in Banach space theory. However, the applications of the classical results are limited by the fact that they force the perturbed sequence to be equivalent to the given sequence. We will develop a more general perturbation theory that does not force equivalence of the sequences. Category:42C15

6. CMB 2007 (vol 50 pp. 85)

Han, Deguang
 Classification of Finite Group-Frames and Super-Frames Given a finite group $G$, we examine the classification of all frame representations of $G$ and the classification of all $G$-frames, \emph{i.e.,} frames induced by group representations of $G$. We show that the exact number of equivalence classes of $G$-frames and the exact number of frame representations can be explicitly calculated. We also discuss how to calculate the largest number $L$ such that there exists an $L$-tuple of strongly disjoint $G$-frames. Keywords:frames, group-frames, frame representations, disjoint framesCategories:42C15, 46C05, 47B10

7. CMB 1999 (vol 42 pp. 37)

Christensen, Ole
 Operators with Closed Range, Pseudo-Inverses, and Perturbation of Frames for a Subspace Recent work of Ding and Huang shows that if we perturb a bounded operator (between Hilbert spaces) which has closed range, then the perturbed operator again has closed range. We extend this result by introducing a weaker perturbation condition, and our result is then used to prove a theorem about the stability of frames for a subspace. Category:42C15

8. CMB 1998 (vol 41 pp. 398)

Dziubański, Jacek; Hernández, Eugenio
 Band-limited wavelets with subexponential decay It is well known that the compactly supported wavelets cannot belong to the class $C^\infty({\bf R})\cap L^2({\bf R})$. This is also true for wavelets with exponential decay. We show that one can construct wavelets in the class $C^\infty({\bf R})\cap L^2({\bf R})$ that are almost'' of exponential decay and, moreover, they are band-limited. We do this by showing that we can adapt the construction of the Lemari\'e-Meyer wavelets \cite{LM} that is found in \cite{BSW} so that we obtain band-limited, $C^\infty$-wavelets on $\bf R$ that have subexponential decay, that is, for every $0<\varepsilon<1$, there exits $C_\varepsilon>0$ such that $|\psi(x)|\leq C_\varepsilon e^{-|x|^{1-\varepsilon}}$, $x\in\bf R$. Moreover, all of its derivatives have also subexponential decay. The proof is constructive and uses the Gevrey classes of functions. Keywords:Wavelet, Gevrey classes, subexponential decayCategory:42C15
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