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1.
We prove that for any expansive n×n integral matrix A with |det A|=2, there exist A-dilation minimally supported frequency
(MSF) wavelets that are associated with a multiresolution analysis (MRA). The condition |det A|=2 was known to be necessary,
and we prove that it is sufficient. A wavelet set is the support set of the Fourier transform of an MSF wavelet. We give some
concrete examples of MRA wavelet sets in the plane. The same technique of proof is also applied to yield an existence result
for A-dilation MRA subspace wavelets. 相似文献
2.
In this paper we study properties of generalized multiresolution analyses (GMRAs) and wavelets associated with rational dilations.
We characterize the class of GMRAs associated with rationally dilated wavelets extending the result of Baggett, Medina, and
Merrill. As a consequence, we introduce and derive the properties of the dimension function of rationally dilated wavelets.
In particular, we show that any mildly regular wavelet must necessarily come from an MRA (possibly of higher multiplicity)
extending Auscher’s result from the setting of integer dilations to that of rational dilations. We also characterize all 3
interval wavelet sets for all positive dilation factors. Finally, we give an example of a rationally dilated wavelet dimension
function for which the conventional algorithm for constructing integer dilated wavelet sets fails. 相似文献
3.
We introduce the concepts of quasi-biorthogonal frame multiresolution analyses and quasi-biorthogonal frame wavelets which are natural generalizations of biorthogonal multiresolution analyses and biorthogonal wavelets, respectively. Necessary and sufficient conditions for quasi-biorthogonal frame multiresolution analyses to admit quasi-biorthogonal wavelet frames are given, and a non-trivial example of quasi-biorthogonal frame multiresolution analyses admitting quasi-biorthogonal frame wavelets is constructed. Finally, we characterize the pair of quasi-biorthogonal frame wavelets that is associated with quasi-biorthogonal frame multiresolution analyses. 相似文献
4.
Rong-Qing Jia 《Advances in Computational Mathematics》2009,30(2):177-200
In this paper we investigate spline wavelets on the interval with homogeneous boundary conditions. Starting with a pair of
families of B-splines on the unit interval, we give a general method to explicitly construct wavelets satisfying the desired
homogeneous boundary conditions. On the basis of a new development of multiresolution analysis, we show that these wavelets
form Riesz bases of certain Sobolev spaces. The wavelet bases investigated in this paper are suitable for numerical solutions
of ordinary and partial differential equations.
Supported in part by NSERC Canada under Grant OGP 121336. 相似文献
5.
A construction of interpolating wavelets on invariant sets 总被引:8,自引:0,他引:8
We introduce the concept of a refinable set relative to a family of contractive mappings on a metric space, and demonstrate how such sets are useful to recursively construct interpolants which have a multiscale structure. The notion of a refinable set parallels that of a refinable function, which is the basis of wavelet construction. The interpolation points we recursively generate from a refinable set by a set-theoretic multiresolution are analogous to multiresolution for functions used in wavelet construction. We then use this recursive structure for the points to construct multiscale interpolants. Several concrete examples of refinable sets which can be used for generating interpolatory wavelets are included.
6.
Zhi Hua Zhang 《数学学报(英文版)》2011,27(1):203-218
The notion of quasi-biorthogonal frame wavelets is a generalization of the notion of orthogonal wavelets. A quasi-biorthogonal
frame wavelet with the cardinality r consists of r pairs of functions. In this paper we first analyze the local property of the quasi-biorthogonal frame wavelet and show that
its each pair of functions generates reconstruction formulas of the corresponding subspaces. Next we show that the lower bound
of its cardinalities depends on a pair of dual frame multiresolution analyses deriving it. Finally, we present a split trick
and show that any quasi-biorthogonal frame wavelet can be split into a new quasi-biorthogonal frame wavelet with an arbitrarily
large cardinality. For generality, we work in the setting of matrix dilations. 相似文献
7.
The paper is concerned with a construction of new spline-wavelet bases on the interval. The resulting bases generate multiresolution
analyses on the unit interval with the desired number of vanishing wavelet moments for primal and dual wavelets. Both primal
and dual wavelets have compact support. Inner wavelets are translated and dilated versions of well-known wavelets designed
by Cohen, Daubechies, and Feauveau. Our objective is to construct interval spline-wavelet bases with the condition number
which is close to the condition number of the spline wavelet bases on the real line, especially in the case of the cubic spline
wavelets. We show that the constructed set of functions is indeed a Riesz basis for the space L
2 ([0, 1]) and for the Sobolev space H
s
([0, 1]) for a certain range of s. Then we adapt the primal bases to the homogeneous Dirichlet boundary conditions of the first order and the dual bases to
the complementary boundary conditions. Quantitative properties of the constructed bases are presented. Finally, we compare
the efficiency of an adaptive wavelet scheme for several spline-wavelet bases and we show a superiority of our construction.
Numerical examples are presented for the one-dimensional and two-dimensional Poisson equations where the solution has steep
gradients. 相似文献
8.
R. A. Zalik 《Applied and Computational Harmonic Analysis》1999,7(3):315
Recently we found a family of nearly orthonormal affine Riesz bases of compact support and arbitrary degrees of smoothness, obtained by perturbing the one-dimensional Haar mother wavelet using B-splines. The mother wavelets thus obtained are symmetric and given in closed form, features which are generally lacking in the orthogonal case. We also showed that for an important subfamily the wavelet coefficients can be calculated in O(n) steps, just as for orthogonal wavelets. It was conjectured by Aldroubi, and independently by the author, that these bases cannot be obtained by a multiresolution analysis. Here we prove this conjecture. The work is divided into four sections. The first section is introductory. The main feature of the second is simple necessary and sufficient conditions for an affine Riesz basis to be generated by a multiresolution analysis, valid for a large class of mother wavelets. In the third section we apply the results of the second section to several examples. In the last section we show that our bases cannot be obtained by a multiresolution analysis. 相似文献
9.
刻画了L~2(R~n)中具有扩展矩阵伸缩的广义低通滤波器和多尺度分析Parseval框架小波(缩写为MRA PFW).首先,研究了伪逆的尺度函数、广义的低通滤波器和MRA PFW,给出它们的一些刻画.接着,我们给出与MRA PFW相联系的几类乘子的一些刻画.最后,给出了一个例子来证明的结论. 相似文献
10.
In this article, we introduce vector-valued multiresolution analysis and the biorthogonal vector-valued wavelets with four-scale. The existence of a class of biorthogonal vector-valued wavelets with compact support associated with a pair of biorthogonal vector-valued scaling functions with compact support is discussed. A method for designing a class of biorthogonal compactly supported vector-valued wavelets with four-scale is proposed by virtue of multiresolution analysis and matrix theory. The biorthogonality properties concerning vector-valued wavelet packets are characterized with the aid of time–frequency analysis method and operator theory. Three biorthogonality formulas regarding them are presented. 相似文献
11.
We introduce the concept of the modular function for a shift-invariant subspace that can be represented by normalized tight frame generators for the shift-invariant subspace and prove that it is independent of the selections of the frame generators for the subspace. We shall apply it to study the connections between the dimension functions of wavelet frames for any expansive integer matrix and the multiplicity functions for general multiresolution analysis (GMRA). Given a frame mutiresolution analysis (FMRA), we show that the standard construction formula for orthonormal multiresolution analysis wavelets does not yield wavelet frames unless the underlying FMRA is an MRA. A modified explicit construction formula for FMRA wavelet frames is given in terms of the frame scaling functions and the low-pass filters.
12.
13.
This paper studiesapproximate multiresolution analysisfor spaces generated by smooth functions providing high-order semi-analytic cubature formulas for multidimensional integral operators of mathematical physics. Since these functions satisfy refinement equations with any prescribed accuracy, methods from wavelet theory can be applied. We obtain an approximate decomposition of the finest scale space into almost orthogonal wavelet spaces. For the example of the Gaussian function we study some properties of the analytic prewavelets and describe the projection operators onto the wavelet spaces. The multivariate wavelets retain the property of the scaling function to provide efficient analytic expressions for the action of important integral operators, which leads to sparse and semi-analytic representations of these operators. 相似文献
14.
A multiresolution analysis for a Hilbert space realizes the Hilbert space as the direct limit of an increasing sequence of closed subspaces. In a previous paper, we showed how, conversely, direct limits could be used to construct Hilbert spaces which have multiresolution analyses with desired properties. In this paper, we use direct limits, and in particular the universal property which characterizes them, to construct wavelet bases in a variety of concrete Hilbert spaces of functions. Our results apply to the classical situation involving dilation matrices on L2(Rn), the wavelets on fractals studied by Dutkay and Jorgensen, and Hilbert spaces of functions on solenoids. 相似文献
15.
Orthogonal and Nonorthogonal Multiresolution Analysis, Scale Discrete and Exact Fully Discrete Wavelet Transform on the Sphere 总被引:1,自引:0,他引:1
Based on a new definition of dilation a scale discrete version of spherical multiresolution is described, starting from a
scale discrete wavelet transform on the sphere. Depending on the type of application, different families of wavelets are chosen.
In particular, spherical Shannon wavelets are constructed that form an orthogonal multiresolution analysis. Finally fully
discrete wavelet approximation is discussed in the case of band-limited wavelets.
June 18, 1996. Date revised: January 14, 1997. 相似文献
16.
A general approach based on polyphase splines, with analysis in the frequency domain, is developed for studying wavelet frames of periodic functions of one or higher dimensions. Characterizations of frames for shift-invariant subspaces of periodic functions and results on the structure of these subspaces are obtained. Starting from any multiresolution analysis, a constructive proof is provided for the existence of a normalized tight wavelet frame. The construction gives the minimum number of wavelets required. As an illustration of the approach developed, the one-dimensional dyadic case is further discussed in detail, concluding with a concrete example of trigonometric polynomial wavelet frames. 相似文献
17.
Marianna Pensky Brani Vidakovic 《Annals of the Institute of Statistical Mathematics》2001,53(4):681-690
Wavelet-based regression analysis is widely used mostly for equally-spaced designs. For such designs wavelets are superior to other traditional orthonormal bases because of their versatility and ability to parsimoniously describe irregular functions. If the regression design is random, an automatic solution is not available. For such non equispaced designs we propose an estimator that is a projection onto a multiresolution subspace in an associated multiresolution analysis. For defining scaling empirical coefficients in the proposed wavelet series estimator our method utilizes a probabilistic model on the design of independent variables. The paper deals with theoretical aspects of the estimator, in particular MSE convergence rates. 相似文献
18.
19.
Ildar Khalidov Michael Unser John Paul Ward 《Journal of Fourier Analysis and Applications》2013,19(6):1294-1322
The connection between derivative operators and wavelets is well known. Here we generalize the concept by constructing multiresolution approximations and wavelet basis functions that act like Fourier multiplier operators. This construction follows from a stochastic model: signals are tempered distributions such that the application of a whitening (differential) operator results in a realization of a sparse white noise. Using wavelets constructed from these operators, the sparsity of the white noise can be inherited by the wavelet coefficients. In this paper, we specify such wavelets in full generality and determine their properties in terms of the underlying operator. 相似文献
20.
Zhihua Zhang 《印度理论与应用数学杂志》2014,45(2):199-212
A function is called a wavelet if its integral translations and dyadic dilations form an orthonormal basis for L 2(?). The support of the Fourier transform of a wavelet is called its frequency band. In this paper, we study the relation between diameters and measures of frequency bands of wavelets, precisely say, we study the ratio of the measure to the diameter. This reflects the average density of the frequency band of a wavelet. In particular, for multiresolution analysis (MRA) wavelets, we do further research. First, we discuss the relation between diameters and measures of frequency bands of scaling functions. Next, we discuss the relation between frequency bands of wavelets and the corresponding scaling functions. Finally, we give the precise estimate of the measure of frequency bands of wavelets. At the same time, we find that when the diameters of frequency bands tend to infinity, the average densities tend to zero. 相似文献