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1.
In this paper, we introduce a class of vector-valued wavelet packets of space L2(R2,Cκ), which are generalizations of multivariate wavelet packets. A procedure for constructing a class of biorthogonal vector-valued wavelet packets in higher dimensions is presented and their biorthogonality properties are characterized by virtue of matrix theory, time–frequency analysis method, and operator theory. Three biorthogonality formulas regarding these wavelet packets are derived. Moreover, it is shown how to gain new Riesz bases of space L2(R2,Cκ) from these wavelet packets. Relation to some physical theories such as the Higgs field is also discussed. 相似文献
2.
R.S. Pathak 《Applicable analysis》2013,92(5):1068-1084
Wavelet packets in Sobolev space Hs (?) are constructed and their orthogonal properties are derived. Using convolution transform theory, boundedness results for the wavelet packets are obtained in the Bp, ? (?) space. Examples of wavelet packets in Sobolev space are given. 相似文献
3.
We consider a family of basic nonstationary wavelet packets generated using the Haar filters except for a finite number of scales where we allow the use of arbitrary filters. Such a system, which we call a system of Walsh-type wavelet packets, can be considered as a smooth generalization of the Walsh functions. We show that the basic Walsh-type wavelet packets share a number of metric properties with the Walsh system. We prove that the system constitutes a Schauder basis for Lp(
), 1<p<∞, and we construct an explicit function in L1(
) for which the expansion fails. Then we prove that expansions of Lp(
)-functions, 1<p<∞, in the Walsh-type wavelet packets converge pointwise a.e. Finally, we prove that the analogous results are true for periodic Walsh-type wavelet packets in Lp[0,1). 相似文献
4.
Biswaranjan Behera 《Proceedings Mathematical Sciences》2001,111(4):439-463
The orthonormal basis generated by a wavelet ofL
2(ℝ) has poor frequency localization. To overcome this disadvantage Coifman, Meyer, and Wickerhauser constructed wavelet packets.
We extend this concept to the higher dimensions where we consider arbitrary dilation matrices. The resulting basis ofL
2(ℝ
d
) is called the multiwavelet packet basis. The concept of wavelet frame packet is also generalized to this setting. Further,
we show how to construct various orthonormal bases ofL
2(ℝ
d
) from the multiwavelet packets. 相似文献
5.
This article obtains the nonseparable version of wavelet packets onℝ
d and generalizes the “unstability” result of nonorthogonal wavelet packets in Cohen-Daubechies to higher dimensional cases.
Professor Ruilin Long died on August 13, 1996. 相似文献
6.
Sandra Saliani 《Constructive Approximation》2011,33(1):15-39
Wavelet packets provide an algorithm with many applications in signal processing together with a large class of orthonormal
bases of L
2(ℝ), each one corresponding to a different splitting of L
2(ℝ) into a direct sum of its closed subspaces. The definition of wavelet packets is due to the work of Coifman, Meyer, and
Wickerhauser, as a generalization of the Walsh system. A question has been posed since then: one asks if a (general) wavelet
packet system can be an orthonormal basis for L
2(ℝ) whenever a certain set linked to the system, called the “exceptional set” has zero Lebesgue measure. This answer to this
question affects the quality of wavelet packet approximation. In this paper we show that the answer to this question is negative
by providing an explicit example. In the proof we make use of the “local trace function” by Dutkay and the generalized shift-invariant
system machinery developed by Ron and Shen. 相似文献
7.
Morten Nielsen 《逼近论及其应用》2002,18(1):34-50
It is well known that the-Walsh-Fourier expansion of a function from the block spaceB q([0,1]), 1
B q in certain periodized smooth periodic non-stationary wavelet packets bases based on the Haar filters. We also consider wavelet packets based on the Shannon filters and show that the expansion of Lp-functions, 1相似文献
8.
Qing-jiang Chen Jin-shun Feng Zheng-xing Cheng 《Journal of Applied Mathematics and Computing》2006,22(3):41-53
In the paper matrix-valued multiresolution analysis and matrix-valued wavelet packets of spaceL 2(R n ,C s x s) are introduced. A procedure for constructing a class of matrix-valued wavelet packets in higher dimensions is proposed. The properties for the matrix-valued multivariate wavelet packets are investigated by using integral transform, algebra theory and operator theory. Finally, a new orthonormal basis ofL 2(R n ,C s x s) is derived from the orthogonal multivariate matrix-valued wavelet packets. 相似文献
9.
Biswaranjan Behera 《Journal of Mathematical Analysis and Applications》2007,328(2):1237-1246
A multiresolution analysis was defined by Gabardo and Nashed for which the translation set is a discrete set which is not a group. We construct the associated wavelet packets for such an MRA. Further, from the collection of dilations and translations of the wavelet packets, we characterize the subcollections which form orthonormal bases for L2(R). 相似文献
10.
Nonorthogonal Wavelet Packets with r Scaling Functions 总被引:1,自引:0,他引:1
In this paper, we discuss the multiresolution analysis generated by finite scaling functions 1, 2,... , , r in L2(R). We also consider the direct wavelet decomposition and direct wavelet packet decomposition in L2(R). Besides, we obtain some results about (dual) nonorthogonal wavelet packet as well as the stability of the basis generating the packet. At last, we give an example of nonorthogonal wavelet packets of multiplicity r by using fractal interpolation functions. 相似文献
11.
In this paper, the concept of vector-valued wavelet packets in space L 2(?+, ? N ) is introduced. Some properties of vector-valued wavelets packets are studied and orthogonality formulas of these wavelets packets are obtained. New orthonormal basis of L 2(?+, ? N ) is obtained by constructing a series of subspaces of vector-valued wavelet packets. 相似文献
12.
对具有任意伸缩矩阵A的插值加细函数,给出对应于L^2(R^s)中的小波包的一个构造方法.采样空间被直接分解来取代对加细函数的符号分解.按照这个方法构造的插值小波包能对基插值空间提供较为精细的分解,因而对自适应的插值给出较好的局部化. 相似文献
13.
《Chaos, solitons, and fractals》2007,31(4):1024-1034
In this paper, vector-valued multiresolution analysis and orthogonal vector-valued wavelets are introduced. The definition for orthogonal vector-valued wavelet packets is proposed. A necessary and sufficient condition on the existence of orthogonal vector-valued wavelets is derived by means of paraunitary vector filter bank theory. An algorithm for constructing a class of compactly supported orthogonal vector-valued wavelets is presented. The properties of the vector-valued wavelet packets are investigated by using operator theory and algebra theory. In particular, it is shown how to construct various orthonormal bases of L2(R, Cs) from the orthogonal vector-valued wavelet packets. 相似文献
14.
15.
Holger Rauhut 《Advances in Computational Mathematics》2005,22(1):1-20
In spaces of trigonometric polynomials, the minimum of the angular variance is determined, which is a time localization measure forL
2
2
. Wavelets and wavelet packets are constructed with the resulting polynomials. 相似文献
16.
In this paper, the notion of two-direction vector-valued multiresolution analysis and the two-direction orthogonal vector-valued wavelets are introduced. The definition for two-direction orthogonal vector-valued wavelet packets is proposed. An algorithm for constructing a class of two-direction orthogonal vector-valued compactly supported wavelets corresponding to the two-direction orthogonal vector-valued compactly supported scaling functions is proposed by virtue of matrix theory and time-frequency analysis method. The properties of the two-direction vector-valued wavelet packets are investigated. At last, the direct decomposition relation for space L2(R)r is presented. 相似文献
17.
Song LI Guo Mao WANG Zhi Song LIU 《数学学报(英文版)》2005,21(6):1475-1486
The purpose of this paper is to investigate the mean size formula of wavelet packets in Lp for 0 〈 p ≤ ∞. We generalize a mean size formula of wavelet packets given in terms of the p-norm joint spectral radius and we also give some asymptotic formulas for the Lp-norm or quasi-norm on the subdivision trees. All results will be given in the general setting, 相似文献
18.
Sylvain Sardy Andrew G. Bruce Paul Tseng 《Journal of computational and graphical statistics》2013,22(2):361-379
Abstract An important class of nonparametric signal processing methods entails forming a set of predictors from an overcomplete set of basis functions associated with a fast transform (e.g., wavelet packets). In these methods, the number of basis functions can far exceed the number of sample values in the signal, leading to an ill-posed prediction problem. The “basis pursuit” denoising method of Chen, Donoho, and Saunders regularizes the prediction problem by adding an l 1 penalty term on the coefficients for the basis functions. Use of an l 1 penalty instead of l 2 has significant benefits, including higher resolution of signals close in time/frequency and a more parsimonious representation. The l 1 penalty, however, poses a challenging optimization problem that was solved by Chen, Donoho and Saunders using a novel application of interior-point algorithms (IP). This article investigates an alternative optimization approach based on block coordinate relaxation (BCR) for sets of basis functions that are the finite union of sets of orthonormal basis functions (e.g., wavelet packets). We show that the BCR algorithm is globally convergent, and empirically, the BCR algorithm is faster than the IP algorithm for a variety of signal denoising problems. 相似文献
19.
The problem of interpolation on the unit sphere S
d
by spherical polynomials of degree at most n is shown to be related to the interpolation on the unit ball B
d
by polynomials of degree n. As a consequence several explicit sets of points on S
d
are given for which the interpolation by spherical polynomials has a unique solution. We also discuss interpolation on the unit disc of R
2 for which points are located on the circles and each circle has an even number of points. The problem is shown to be related to interpolation on the triangle in a natural way. 相似文献
20.
L. F. Villemoes 《Constructive Approximation》1997,13(3):329-355
We consider the approximation in L
2
R of a given function using finite linear combinations of Walsh atoms, which are Walsh functions localized to dyadic intervals,
also called Haar—Walsh wavelet packets. It is shown that up to a constant factor, a linear combination of K atoms can be represented to relative error ɛ by a linear combination of orthogonal atoms.
In finite dimension N, best approximation with K orthogonal atoms can be realized with an algorithm of order . A faster algorithm of order solves the problem with indirect control over K. Therefore the above result connects algorithmic and theoretical best approximation.
Date received: July 6, 1995. Date revised: January 8, 1996. 相似文献