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1.
In this paper we consider the Dunkl operators T
j
, j = 1, . . . , d, on and the harmonic analysis associated with these operators. We define a continuous Dunkl Gabor transform, involving the Dunkl
translation operator, by proceeding as mentioned in [20] by C.Wojciech and G. Gigante. We prove a Plancherel formula, an inversion formula and a weak uncertainty principle for it. Then, we show that the portion of the continuous Dunkl Gabor transform
lying outside some set of finite measure cannot be arbitrarily too small. Similarly, using the basic theory for the Dunkl
continuous wavelet transform introduced by K. Trimèche in [18], an analogous of this result for the Dunkl continuous wavelet
transform is given. Finally, an analogous of Heisenberg’s inequality for a continuous Dunkl Gabor transform (resp. Dunkl continuous
wavelet transform) is proved.
相似文献
2.
We demonstrate that the Plancherel transform for Type-I groups provides
one with a natural, unified perspective for the generalized continuous wavelet
transform, on the one hand, and for a class of Wigner functions, on the other. We
first prove that a Plancherel inversion formula, well known for Bruhat functions on
the group, holds for a much larger class of functions. This result allows us to view
the wavelet transform as essentially the inverse Plancherel transform. The wavelet
transform of a signal is an L2-function on an appropriately chosen group while
the Wigner function is defined on a coadjoint orbit of the group and serves as an
alternative characterization of the signal, which is often used in practical applications.
The Plancherel transform maps L2-functions on a group unitarily to fields
of Hilbert-Schmidt operators, indexed by unitary irreducible representations of the
group. The wavelet transform can essentially be looked upon as a restricted inverse
Plancherel transform, while Wigner functions are modified Fourier transforms of
inverse Plancherel transforms, usually restricted to a subset of the unitary dual of
the group. Some known results on both Wigner functions and wavelet transforms,
appearing in the literature from very different perspectives, are naturally unified
within our approach. Explicit computations on a number of groups illustrate the
theory.
Communicated by Gian Michele Graf
submitted 05/06/01, accepted: 19/09/02 相似文献
3.
Jakob Lemvig 《Advances in Computational Mathematics》2009,30(3):231-247
For sufficiently small translation parameters, we prove that any bandlimited function ψ, for which the dilations of its Fourier transform form a partition of unity, generates a wavelet frame with a dual frame
also having the wavelet structure. This dual frame is generated by a finite linear combination of dilations of ψ with explicitly given coefficients. The result allows a simple construction procedure for pairs of dual wavelet frames whose
generators have compact support in the Fourier domain and desired time localization. The construction is based on characterizing
equations for dual wavelet frames and relies on a technical condition. We exhibit a general class of function satisfying this
condition; in particular, we construct piecewise polynomial functions satisfying the condition.
相似文献
4.
The homogeneous approximation property (HAP) for frames is useful in practice and has been developed recently. In this paper, we study the HAP for the continuous wavelet transform. We show that every pair of admissible wavelets possesses the HAP in L2 sense, while it is not true in general whenever pointwise convergence is considered. We give necessary and sufficient conditions for the pointwise HAP to hold, which depends on both wavelets and functions to be reconstructed. 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.
H. Lamouchi 《Integral Transforms and Special Functions》2016,27(1):43-54
We use some estimating of orthogonal projection in a reproducing kernel Hilbert space, to prove a sharp quantitaive form of Shapiro's mean dispersion theroem with generalized dispersion for the short time Fourier transform. Other forms of localization of orthonormal sequences in L2?d) notably the umbrella theorem, are also proved for the short time Fourier transform. 相似文献
7.
Complex Wavelets for Shift Invariant Analysis and Filtering of Signals 总被引:14,自引:0,他引:14
Nick Kingsbury 《Applied and Computational Harmonic Analysis》2001,10(3):283
This paper describes a form of discrete wavelet transform, which generates complex coefficients by using a dual tree of wavelet filters to obtain their real and imaginary parts. This introduces limited redundancy (2m:1 for m-dimensional signals) and allows the transform to provide approximate shift invariance and directionally selective filters (properties lacking in the traditional wavelet transform) while preserving the usual properties of perfect reconstruction and computational efficiency with good well-balanced frequency responses. Here we analyze why the new transform can be designed to be shift invariant and describe how to estimate the accuracy of this approximation and design suitable filters to achieve this. We discuss two different variants of the new transform, based on odd/even and quarter-sample shift (Q-shift) filters, respectively. We then describe briefly how the dual tree may be extended for images and other multi-dimensional signals, and finally summarize a range of applications of the transform that take advantage of its unique properties. 相似文献
8.
Andreas Rieder 《Numerische Mathematik》1990,58(1):875-894
Summary We extend the continuous wavelet transform to Sobolev spacesH
s() for arbitrary reals and show that the transformed distribution lies in the fiber spaces
. This generalisation of the wavelet transform naturally leads to a unitary operator between these spaces.Further the asymptotic behaviour of the transforms ofL
2-functions for small scaling parameters is examined. In special cases the wevelet transform converges to a generalized derivative of its argument. We also discuss the consequences for the discrete wavelet transform arising from this property. Numerical examples illustrate the main result.Supported by the Deutsche Forschungsgemeinschaft under grant Lo 310/2-4 相似文献
9.
The paper at hand is concerned with creating a flexible wavelet theory on the three sphere S3 and the rotation group SO(3). The theory of zonal functions and reproducing kernels will be used to develop conditions for an admissible wavelet. After explaining some preliminaries on group actions and some basics on approximation theory, we will prove reconstruction formulas of linear and bilinear wavelet transformed L2‐functions on S3. Moreover, specific examples will be constructed and visualized. Second, we deal with the construction of wavelets on the rotation group SO(3). It will be shown that the Radon transform of a wavelet packet on SO(3) gives a wavelet packet on S2 for every fixed detection direction. Copyright © 2010 John Wiley & Sons, Ltd. 相似文献
10.
Boris Rubin 《Journal of Fourier Analysis and Applications》1998,4(2):175-197
The generalized Calderón reproducing formula involving “wavelet measure” is established for functions f ∈ Lp(ℝn). The special choice of the wavelet measure in the reproducing formula gives rise to the continuous decomposition of f into
wavelets, and enables one to obtain inversion formulae for generalized windowed X-ray transforms, the Radon transform, and
k-plane transforms. The admissibility conditions for the wavelet measure μ are presented in terms of μ itself and in terms
of the Fourier transform of μ.
Acknowledgements and Notes. Partially sponsored by the Edmund Landau Center for research in Mathematical Analysis, supported by the Minerva Foundation
(Germany). 相似文献
11.
Eugen J. Ionascu David R. Larson Carl M. Pearcy 《Journal of Fourier Analysis and Applications》1998,4(6):711-721
It is proved that associated with every wavelet set is a closely related “regularized” wavelet set which has very nice properties.
Then it is shown that for many (and perhaps all) pairs E, F, of wavelet sets, the corresponding MSF wavelets can be connected
by a continuous path in L2(ℝ) of MSF wavelets for which the Fourier transform has support contained in E ∪ F. Our technique applies, in particular,
to the Shannon and Journe wavelet sets. 相似文献
12.
We introduce a new method to construct large classes of minimally supported frequency (MSF) wavelets of the Hardy space H
2
(ℝ)and symmetric MSF wavelets of L
2
(ℝ),and discuss the classification of such wavelets. As an application, we show that there are uncountably many such wavelet sets
of L
2
(ℝ)and H
2
(ℝ).We also enumerate some of the symmetric wavelet sets of L
2
(ℝ)and all wavelet sets of H
2
(ℝ)consisting of three intervals. Finally, we construct families of MSF wavelets of L
2
(ℝ)with Fourier transform even and not vanishing in any neighborhood of the origin. 相似文献
13.
Emna Tefjeni 《Integral Transforms and Special Functions》2020,31(8):669-684
ABSTRACT In this paper, we present some new elements of harmonic analysis related to the right-sided multivariate continuous quaternion wavelet transform. The main objective of this article is to introduce the concept of the right-sided multivariate continuous quaternion wavelet transform and investigate its different properties using the machinery of multivariate quaternion Fourier transform. Last, we have proven a number of uncertainty principles for the right-sided multivariate continuous quaternion wavelet transform. 相似文献
14.
Factoring wavelet transforms into lifting steps 总被引:236,自引:0,他引:236
This article is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with
finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are
also known as ladder structures. This decomposition corresponds to a factorization of the polyphase matrix of the wavelet
or subband filters into elementary matrices. That such a factorization is possible is well-known to algebraists (and expressed
by the formulaSL(n;R[z, z−1])=E(n;R[z, z−1])); it is also used in linear systems theory in the electrical engineering community. We present here a self-contained derivation,
building the decomposition from basic principles such as the Euclidean algorithm, with a focus on applying it to wavelet filtering.
This factorization provides an alternative for the lattice factorization, with the advantage that it can also be used in the
biorthogonal, i.e., non-unitary case. Like the lattice factorization, the decomposition presented here asymptotically reduces
the computational complexity of the transform by a factor two. It has other applications, such as the possibility of defining
a wavelet-like transform that maps integers to integers.
Research Tutorial
Acknowledgements and Notes. Page 264. 相似文献
15.
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
16.
D. E. Ferreyra 《Numerical Functional Analysis & Optimization》2017,38(6):770-798
We study the behavior of the best simultaneous approximation to two functions from a convex set in Lp spaces, 2<p<∞, on a finite union of intervals when its measure tends to zero. In particular, we give su?cient conditions over the differentiability of two functions to assure existence of the best simultaneous local approximation from the class of algebraic polynomials of a fixed degree. These conditions are weaker than the ordinary differentiability given in previous works. More precisely, we consider differentiable functions in the sense Lp. 相似文献
17.
In this article we use the C 1 wavelet bases on Powell-Sabin triangulations to approximate the solution of the Neumann problem for partial differential equations. The C 1 wavelet bases are stable and have explicit expressions on a three-direction mesh. Consequently, we can approximate the solution of the Neumann problem accurately and stably. The convergence and error estimates of the numerical solutions are given. The computational results of a numerical example show that our wavelet method is well suitable to the Neumann boundary problem. 相似文献
18.
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
19.
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
20.
In this article, we study the behavior of best simultaneous L p -approximation by algebraic polynomials on a union of intervals when the measure of them tend to zero. We also get an interpolation result. 相似文献