Riesz Bases and Multiresolution Analyses

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.     

Tight Frame Wavelets, their Dimension Functions, MRA Tight Frame Wavelets and Connectivity Properties

This is a continuation of our study of generalized low pass filters and MRA frame wavelets. In this first study we concentrated on the construction of such functions. Here we are particularly interested in the role played by the dimension function. In particular we characterize all semi-orthogonal Tight Frame Wavelets (TFW) by showing that they correspond precisely to those for which the dimension function is non-negative integer-valued. We also show that a TFW arises from our MRA construction if and only if the dimension of a particular linear space is either zero or one. We present many examples. In addition we obtain a result concerning the connectivity of TFW's that are MSF tight frame wavelets.     

Symmetries in Projective Multiresolution Analyses

We give an equivariant version of Packer and Rieffel’s theorem on sufficient conditions for the existence of orthonormal wavelets in projective multiresolution analysis. Suppose that the scaling functions are invariant with respect to some finite group action. We give sufficient conditions for the existence of wavelets with similar invariance. Research supported in part by the Research Council of Norway, project number NFR 154077/420. Some of the final work was also done with the support from the project NFR 170620/V30.     

Frames of p-adic wavelets and orbits of the affine group

The general construction of frames of p-adic wavelets is described. We consider the orbit of a generic mean zero locally constant function with compact support (mean zero test function) with respect to the action of the p-adic affine group and show that this orbit is a uniform tight frame. We discuss the relations of this result with the multiresolution wavelet analysis. The text was submitted by the authors in English.     

具有特殊伸缩矩阵的Parseval框架小波集的结构

揭示具有特殊伸缩矩阵的Parseval框架小波集的丰富结构.借助于平移不变空间和维数函数,研究了具有特殊伸缩矩阵M的Parseval框架小波(M-PFW)、半正交M-PFW和MRA M-PFW的各种性质,探讨了M-PFW集合的各种子类,给出了这些子类的构造性算例.     

Minimally Supported Frequency Composite Dilation Parseval Frame Wavelets

A composite dilation Parseval frame wavelet is a collection of functions generating a Parseval frame for L ²(ℝ ⁿ ) under the actions of translations from a full rank lattice and dilations by products of elements of groups A and B. A minimally supported frequency composite dilation Parseval frame wavelet has generating functions whose Fourier transforms are characteristic functions of sets contained in a lattice tiling set. Constructive proofs are used to establish the existence of minimally supported frequency composite dilation Parseval frame wavelets in arbitrary dimension using any finite group B, any full rank lattice, and an expanding matrix generating the group A and normalizing the group B. Moreover, every such system is derived from a Parseval frame multiresolution analysis. Multiple examples are provided including examples that capture directional information.      

Wavelets with Frame Multiresolution Analysis

A frame multiresolution (FMRA for short) orthogonalwavelet is a single-function orthogonal wavelet such that theassociated scaling space V0 admits a normalized tight frame(under translations). In this article, we prove that for anyexpansive matrix A with integer entries, there existA-dilation FMRA orthogonal wavelets. FMRA orthogonal waveletsfor some other expansive matrix with non integer entries are also discussed.     

Wavelets Based on Orthogonal Polynomials

We present a unified approach for the construction of polynomial wavelets. Our main tool is orthogonal polynomials. With the help of their properties we devise schemes for the construction of time localized polynomial bases on bounded and unbounded subsets of the real line. Several examples illustrate the new approach.

     

一类带限的插值小波

Ｓｈａｎｎｏｎ尺度函数具有带限性质，正交性质、取样性质，但它不在Ｌ（Ｒ）中，本文引入小波，它的尺度函数不仅具有上述性质而且在Ｌ（Ｒ）中，甚至更多。     

一对拟双正交框架小波

对于一对对偶框架多尺度分析,借助于滤波器,我们构造一对对偶框架小波和一对拟双正交框架小波,并且指出一对对偶框架小波和一对拟双正交框架小波的滤波器所满足的充分必要条件.     

The wavelet dimension function is the trace function of a shift-invariant system

In this note, we observe that the dimension function associated with a wavelet system is the trace of the Gramian fibers of the shift-invariant system generated by the negative dilations of the mother wavelets. When this shift-invariant system is a tight frame, each of the Gramian fibers is an orthogonal projector, and its trace, then, coincides with its rank. This connection leads to simple proofs of several results concerning the dimension function, and the arguments extend to the bi-frame case.

     

Tight compactly supported wavelet frames of arbitrarily high smoothness

Based on Ron and Shen's new method for constructing tight wave-let frames, we show that one can construct, for any dilation matrix, and in any spatial dimension, tight wavelet frames generated by compactly supported functions with arbitrarily high smoothness.

     

小波分析及信源熵在股市预测分析中的应用

提出了一种预测股市行情的新指标——信源熵 ,结合小波分析和 MATLAB工具来分析预测股票的未来走势 .并用实例说明了信源熵和小波分析在预测股票的走势上有很大优越性和准确性 .     

A Characterization of Generalized Frame MRAs Deriving Orthonormal Wavelets

In 2000, Papadakis announced that any orthonormal wavelet must be derived by a generalized frame MRA （GFMRA）. In this paper, we give a characterization of GFMRAs which can derive orthonormal wavelets, and show a general approach to the constructions of non-MRA wavelets. Finally we present two examples to illustrate the theory.     

关于Parseval框架小波的刻画

In this paper,we characterize all generalized low pass filters and MRA Parseval frame wavelets in L 2 （R n ） with matrix dilations of the form （Df）（x） =√ 2f（Ax）,where A is an arbitrary expanding n × n matrix with integer coefficients,such that ｜det A｜ = 2.We study the pseudo-scaling functions,generalized low pass filters and MRA Parseval frame wavelets and give some important characterizations about them.Furthermore,we give a characterization of the semiorthogonal MRA Parseval frame wavelets and provide several examples to verify our results.     

一类紧支撑矩阵值正交小波的构造

In this paper,we introduce matrix-valued multiresolution analysis and orthogonal matrix-valued wavelets.We obtain a necessary and sufficient condition on the existence of orthogonal matrix-valued wavelets by means of paraunitary vector filter bank theory.A method for constructing a class of compactly supported orthogonal matrix-valued wavelets is proposed by using multiresolution analysis method and matrix theory.     

向量值正交小波的构造与向量值小波包的特征

The notion of vector-valued multiresolution analysis is introduced and the concept of orthogonal vector-valued wavelets with 3-scale is proposed.A necessary and sufficient condition on the existence of orthogonal vector-valued wavelets is given by means of paraunitary vector filter bank theory.An algorithm for constructing a class of compactly supported orthogonal vector-valued wavelets is presented.Their characteristics is discussed by virtue of operator theory,time-frequency method.Moreover,it is shown how to design various orthonormal bases of space L2(R,Cn) from these wavelet packets.     

Compactly supported tight affine spline frames in

The theory of fiberization is applied to yield compactly supported tight affine frames (wavelets) in from box splines. The wavelets obtained are smooth piecewise-polynomials on a simple mesh; furthermore, they exhibit a wealth of symmetries, and have a relatively small support. The number of ``mother wavelets', however, increases with the increase of the required smoothness.

Two bivariate constructions, of potential practical value, are highlighted. In both, the wavelets are derived from four-direction mesh box splines that are refinable with respect to the dilation matrix .