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1.
We deduce the exact Jackson-type inequalities for the approximations of functions ƒ ∈ L
2(ℝ) in L
2(ℝ) by using partial sums of wavelet series in the cases of Meyer and Shannon-Kotelnikov wavelets.
Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 8, pp. 1119–1127, August, 2008. 相似文献
2.
Maria Skopina 《Journal of Approximation Theory》2000,104(2):361
We investigate expansions of periodic functions with respect to wavelet bases. Direct and inverse theorems for wavelet approximation in C and Lp norms are proved. For the functions possessing local regularity we study the rate of pointwise convergence of wavelet Fourier series. We also define and investigate the “discreet wavelet Fourier transform” (DWFT) for periodic wavelets generated by a compactly supported scaling function. The DWFT has one important advantage for numerical problems compared with the corresponding wavelet Fourier coefficients: while fast computational algorithms for wavelet Fourier coefficients are recursive, DWFTs can be computed by explicit formulas without any recursion and the computation is fast enough. 相似文献
3.
We introduce a new form of nonlinear approximation called restricted approximation . It is a generalization of n -term wavelet approximation in which a weight function is used to control the terms in the wavelet expansion of the approximant.
This form of approximation occurs in statistical estimation and in the characterization of interpolation spaces for certain
pairs of L
p
and Besov spaces. We characterize, both in terms of their wavelet coefficients and also in terms of their smoothness, the
functions which are approximated with a specified rate by restricted approximation. We also show the relation of this form
of approximation with certain types of thresholding of wavelet coefficients.
March 31, 1998. Date accepted: January 28, 1999. 相似文献
4.
The single 2 dilation wavelet multipliers in one-dimensional case and single A-dilation (where A is any expansive matrix with integer entries and |detA| = 2) wavelet multipliers in twodimensional case were completely characterized by Wutam Consortium (1998) and Li Z., et al.
(2010). But there exist no results on multivariate wavelet multipliers corresponding to integer expansive dilation matrix
with the absolute value of determinant not 2 in L
2(ℝ2). In this paper, we choose $2I_2 = \left( {{*{20}c}
2 & 0 \\
0 & 2 \\
} \right)$2I_2 = \left( {\begin{array}{*{20}c}
2 & 0 \\
0 & 2 \\
\end{array} } \right) as the dilation matrix and consider the 2I
2-dilation multivariate wavelet Φ = {ψ
1, ψ
2, ψ
3}(which is called a dyadic bivariate wavelet) multipliers. Here we call a measurable function family f = {f
1, f
2, f
3} a dyadic bivariate wavelet multiplier if Y1 = { F - 1 ( f1 [^(y1 )] ),F - 1 ( f2 [^(y2 )] ),F - 1 ( f3 [^(y3 )] ) }\Psi _1 = \left\{ {\mathcal{F}^{ - 1} \left( {f_1 \widehat{\psi _1 }} \right),\mathcal{F}^{ - 1} \left( {f_2 \widehat{\psi _2 }} \right),\mathcal{F}^{ - 1} \left( {f_3 \widehat{\psi _3 }} \right)} \right\} is a dyadic bivariate wavelet for any dyadic bivariate wavelet Φ = {ψ
1, ψ
2, ψ
3}, where [^(f)]\hat f and F
−1 denote the Fourier transform and the inverse transform of function f respectively. We study dyadic bivariate wavelet multipliers, and give some conditions for dyadic bivariate wavelet multipliers.
We also give concrete forms of linear phases of dyadic MRA bivariate wavelets. 相似文献
5.
This paper presents a construction of the n = 2 (mod 4) Clifford algebra Cl
n,0-valued admissible wavelet transform using the admissible similitude group SIM(n), a subgroup of the affine group of
\mathbbRn{\mathbb{R}^{n}} . We express the admissibility condition in terms of the Cl
n,0 Clifford Fourier transform (CFT). We show that its fundamental properties such as inner product, norm relation, and inversion
formula can be established whenever the Clifford admissible wavelet satisfies a particular admissibility condition. As an
application we derive a Heisenberg type uncertainty principle for the Clifford algebra Cl
n,0-valued admissible wavelet transform. Finally, we provide some basic examples of these extended wavelets such as Clifford
Morlet wavelets and Clifford Hermite wavelets. 相似文献
6.
S. Albeverio S. Evdokimov M. Skopina 《Proceedings of the Steklov Institute of Mathematics》2009,265(1):1-12
A method for constructing MRA-based p-adic wavelet systems that form Riesz bases in L
2(ℚ
p
) is developed. The method is implemented for an infinite family of MRAs. 相似文献
7.
Reinhard Hochmuth 《Mathematische Nachrichten》2002,244(1):131-149
In L2(0, 1)2) infinitely many different biorthogonal wavelet bases may be introduced by taking tensor products of one–dimensional biorthogonal wavelet bases on the interval (0, 1). Most well–known are the standard tensor product bases and the hyperbolic bases. In [23, 24] further biorthogonal wavelet bases are introduced, which provide wavelet characterizations for functions in anisotropic Besov spaces. Here we address the following question: Which of those biorthogonal tensor product wavelet bases is the most appropriate one for approximating nonlinearly functions from anisotropic Besov spaces? It turns out, that the hyperbolic bases lead to nonlinear algorithms which converge as fast as the corresponding schemes with respect to specific anisotropy adapted bases. 相似文献
8.
We estimate the truncation error of sampling expansions on translationinvariant spaces, generated by integer translations of a single functionand on wavelet subspaces of L
2(R). As a byproduct of themain result, we get the classical Jagerman's bound for Shannon's samplingexpansions. We also examine this error on certain wavelet sampling expansions. 相似文献
9.
We give a characterization of weighted Hardy spaces H
p
(w), valid for a rather large collection of wavelets, 0 <p ≤ 1,and weights w in the Muckenhoupt class A
∞
We improve the previously known results and adopt a systematic point of view based upon the theory of vector-valued Calderón-Zygmund
operators. Some consequences of this characterization are also given, like the criterion for a wavelet to give an unconditional
basis and a criterion for membership into the space from the size of the wavelet coefficients. 相似文献
10.
Construction of Wavelet Analysis in the Space of Discrete Splines Using Zak Transform 总被引:4,自引:0,他引:4
Alexander B. Pevnyi Valery A. Zheludev 《Journal of Fourier Analysis and Applications》2002,8(1):59-84
We consider equidistant discrete splines S(j)S(j),
j ? \mathbbZj\in\mathbb{Z}, which may grow as O(|j|s)O(|j|^s) as |j|?¥|j|\to\infty. Such splines present a relevant tool for digital signal processing. The Zak transforms of B-splines yield the integral representation of discrete splines. We define the wavelet space as a weak orthogonal complement of the coarse-grid space in the fine-grid space. We establish the integral representation of the elements of the wavelet space. We define and characterize the wavelets whose shifts form bases of the wavelet space. By this means we design a wide library of bases for the space of discrete-time signals of power growth and construct multiscale representation of this space. We provide formulas for processing such the signals by discrete spline wavelets. Constructed bases are at the same time the Riesz bases for the space l2l_2. 相似文献
11.
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
12.
Sergio Albeverio Sergei V. Kozyrev 《P-Adic Numbers, Ultrametric Analysis, and Applications》2010,2(4):265-277
The approach to p-adic wavelet theory from the point of view of representation theory is discussed. p-Adic wavelet frames can be constructed as orbits of some p-adic groups of transformations. These groups are automorphisms of the tree of balls in the p-adic space. In the present paper we consider deformations of the standard p-adic metric in many dimensions and construct some corresponding groups of transformations. We build several examples of p-adic wavelet bases. We show that the constructed wavelets are eigenvectors of some pseudodifferential operators. 相似文献
13.
Andy C. Yau Xuecheng Tai Michael K. Ng 《Computational Optimization and Applications》2011,50(2):425-444
In this paper, we deal with l
0-norm data fitting and total variation regularization for image compression and denoising. The l
0-norm data fitting is used for measuring the number of non-zero wavelet coefficients to be employed to represent an image.
The regularization term given by the total variation is to recover image edges. Due to intensive numerical computation of
using l
0-norm, it is usually approximated by other functions such as the l
1-norm in many image processing applications. The main goal of this paper is to develop a fast and effective algorithm to solve
the l
0-norm data fitting and total variation minimization problem. Our idea is to apply an alternating minimization technique to
solve this problem, and employ a graph-cuts algorithm to solve the subproblem related to the total variation minimization.
Numerical examples in image compression and denoising are given to demonstrate the effectiveness of the proposed algorithm. 相似文献
14.
The notion of p-adic multiresolution analysis (MRA) is introduced. We discuss a “natural” refinement equation whose solution (a refinable function) is the characteristic function
of the unit disc. This equation reflects the fact that the characteristic function of the unit disc is a sum of p characteristic functions of mutually disjoint discs of radius p
−1. This refinement equation generates a MRA. The case p=2 is studied in detail. Our MRA is a 2-adic analog of the real Haar MRA. But in contrast to the real setting, the refinable
function generating our Haar MRA is 1-periodic, which never holds for real refinable functions. This fact implies that there
exist infinity many different 2-adic orthonormal wavelet bases in ℒ2(ℚ2) generated by the same Haar MRA. All of these new bases are described. We also constructed infinity many different multidimensional 2-adic Haar orthonormal wavelet bases for ℒ2(ℚ2
n
) by means of the tensor product of one-dimensional MRAs. We also study connections between wavelet analysis and spectral
analysis of pseudo-differential operators. A criterion for multidimensional p-adic wavelets to be eigenfunctions for a pseudo-differential operator (in the Lizorkin space) is derived. We proved also
that these wavelets are eigenfunctions of the Taibleson multidimensional fractional operator. These facts create the necessary
prerequisites for intensive using our wavelet bases in applications. Our results related to the pseudo-differential operators
develop the investigations started in Albeverio et al. (J. Fourier Anal. Appl. 12(4):393–425, 2006).
相似文献
15.
Near Best Tree Approximation 总被引:2,自引:0,他引:2
Baraniuk R.G. DeVore R.A. Kyriazis G. Yu X.M. 《Advances in Computational Mathematics》2002,16(4):357-373
Tree approximation is a form of nonlinear wavelet approximation that appears naturally in applications such as image compression and entropy encoding. The distinction between tree approximation and the more familiar n-term wavelet approximation is that the wavelets appearing in the approximant are required to align themselves in a certain connected tree structure. This makes their positions easy to encode. Previous work [4,6] has established upper bounds for the error of tree approximation for certain (Besov) classes of functions. This paper, in contrast, studies tree approximation of individual functions with the aim of characterizing those functions with a prescribed approximation error. We accomplish this in the case that the approximation error is measured in L
2, or in the case p2, in the Besov spaces B
p
0(L
p
), which are close to (but not the same as) L
p
. Our characterization of functions with a prescribed approximation order in these cases is given in terms of a certain maximal function applied to the wavelet coefficients. 相似文献
16.
We characterize the approximation spaces associated with the best n-term approximation in Lp(R) by elements from
a tight wavelet frame associated with a spline scaling function. The approximation spaces are shown to be interpolation spaces between Lp and classical Besov spaces, and the result coincides with the
result for nonlinear approximation with an orthonormal wavelet with the same smoothness as the spline scaling function. We also show that, under certain conditions, the Besov smoothness can be measured in terms of the sparsity of expansions in the wavelet frame, just like the nonredundant wavelet case. However, the characterization now holds even for wavelet frame systems that do not have the usually
required number of vanishing moments, e.g., for systems built through the Unitary Extension Principle, which can have no more than one vanishing moment. Using these results, we describe a fast algorithm that takes as input any function and provides a near sparsest expansion of it in the framelet system as well as approximants that reach the optimal rate of nonlinear approximation. Together with the existence of a fast algorithm, the absence of the need for vanishing moments may have an important qualitative impact for applications to signal compression, as high vanishing moments usually introduce a Gibbs-type phenomenon (or ringing artifacts)in the approximants. 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.
Under suitable conditions, if the scaling functions ?1 and ?2 generate the multiresolutions V (j)(?1) and V (j)(?2), then their convolution ?1*?2also generates a multiresolution V (j)(?1*?2) More over, if p is an appropriate convolution operator from l 2 into itself and if ? is a scaling function generating the multiresolution V (j)(?),then p*?is a scaling function generating the same multiresolution V (j)(?)=V (j)(p*?). Using these two properties, we group the scaling and wavelet functions into equivalent classes and consider various equivalent basis functions of the associated function spaces We use the n-fold convolution product to construct sequences of multiresolution and wavelet spaces V (j)(?n) and W (j)(?n) with increasing regularity. We discuss the link between multiresolution analysis and Shannon's sampling theory. We then show that the interpolating and orthogonal pre- and post-filters associated with the multiresolution sequence V (0)(?n)asymptotically converge to the ideal lowpass filter of Shannon. We also prove that the filters associated with the sequence of wavelet spaces W (0)(?n)convergeto the ideal bandpass filter. Finally, we construct the basic wavelet sequences ψ b nand show that they tend to Gabor functions. Thisprovides wavelets that are nearly time-frequency optimal. The theory is illustrated with the example of polynomial splines. 相似文献
19.
Based on kernel and wavelet estimators of the evolutionary spectrum and cross-spectrum we propose nonlinear wavelet estimators
of the time varying coefficients of a linear system, whose input and output are locally stationary processes, in the sense
of Dahlhaus (1997). We obtain large sample properties of these estimators, present some simulated examples and derive results
on the L
2-risk for the wavelet threshold estimators, assuming that the coefficients belong to some smoothness class.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
20.
An orthogonal basis for the spaceS
r
m
of discrete periodic splines is constructed. The wavelet decomposition of the spaceS
r
m
form=2
t
is obtained using this basis. We derive recurrence formulas for the transformation from the decomposition with respect to
the orthogonal basis to the wavelet decomposition, as well as recurrence formulas for the inverse transformation.
Translated fromMatematicheskie Zametki, Vol. 67, No. 5, pp. 712–720, May, 2000. 相似文献