Multiwavelet packets and frame packets ofL 2(ℝ d )

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.     

Construction of vector valued wavelet packets on ℝ+ using Walsh-Fourier transform

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.     

Parseval Frame Wavelet Multipliers in L2（Rd）

Let A be a d × d real expansive matrix. An A-dilation Parseval frame wavelet is a function ?? ?? L ²(? ^d ), such that the set $ \left\{ {\left| {\det A} \right|^{\frac{n} {2}} \psi \left( {A^n t - \ell } \right):n \in \mathbb{Z},\ell \in \mathbb{Z}^d } \right\} $ forms a Parseval frame for L ²(? ^d ). A measurable function f is called an A-dilation Parseval frame wavelet multiplier if the inverse Fourier transform of d??? is an A-dilation Parseval frame wavelet whenever ?? is an A-dilation Parseval frame wavelet, where ??? denotes the Fourier transform of ??. In this paper, the authors completely characterize all A-dilation Parseval frame wavelet multipliers for any integral expansive matrix A with |det(A)| = 2. As an application, the path-connectivity of the set of all A-dilation Parseval frame wavelets with a frame MRA in L ²(? ^d ) is discussed.     

The properties of orthogonal matrix-valued wavelet packets in higher dimensions

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.     

A study on compactly supported orthogonal vector-valued wavelets and wavelet packets

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, C^s) from the orthogonal vector-valued wavelet packets.     

Interpolatory Wavelet Packets

In this note, we present a construction of interpolatory wavelet packets. Interpolatory wavelet packets provide a finer decomposition of the 2^jth dilate cardinal interpolation space and hence give a better localization for an adaptive interpolation. This can lead to a more efficient compression scheme which, in turn, provides an interpolation algorithm with a smaller set of data for use in applications.     

A study on orthogonal two-direction vector-valued wavelets and two-direction wavelet packets

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.     

The properties of a class of biorthogonal vector-valued nonseparable bivariate wavelet packets

In this paper, we introduce a class of vector-valued wavelet packets of space L²(R²,C^κ), 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,C^κ) from these wavelet packets. Relation to some physical theories such as the Higgs field is also discussed.     

On the holes of a class of bidimensional nonseparable wavelets

Let I be the 2×2 identity matrix, and M a 2×2 dilation matrix with M²=2I. Since one can explicitly construct M-basic wavelets from an MRA related to M, and many applications employ wavelet bases in R², M-wavelets and wavelet frames have been extensively discussed. This paper focuses on dilation matrices M satisfying M²=2I. For any matrix M integrally similar to , an optimal estimate on the boundary of the holes of M-wavelets is obtained. This result tells us the holes cannot be too large. Contrast to this result, when the modulus of the Fourier transform of an M-wavelet is, up to a constant, a characteristic function on some set, a property of this set is obtained, which shows the holes of this kind of wavelets cannot be too small.     

一类二元半正交小波包的构造

本文给出伸缩矩阵行列式为2的一类二元半正交小波包的构造算法.该小波包是以频域给出的,随着用于小波包分裂的滤波器选取的不同会得到L2(R2)中形态各异的Riesz基,这样使得L2(R2)中小波基的选择更灵活.     

On the <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-consistency of wavelet estimators

This paper deals with the L ^p -consistency of wavelet estimators for a density function based on size-biased random samples. More precisely, we firstly show the L ^p -consistency of wavelet estimators for independent and identically distributed random vectors in R ^d . Then a similar result is obtained for negatively associated samples under the additional assumptions d = 1 and the monotonicity of the weight function.     

The Solution of a Problem of Coifman, Meyer, and?Wickerhauser on Wavelet Packets

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.     

Direct limits, multiresolution analyses, and wavelets

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.     

Wavelet packets in Sobolev space

Wavelet packets in Sobolev space H^s (?) are constructed and their orthogonal properties are derived. Using convolution transform theory, boundedness results for the wavelet packets are obtained in the B_{p, ?} (?) space. Examples of wavelet packets in Sobolev space are given.     

An integral transform associated to the Poisson integral and inversion of Flett potentials

We introduce a new weighted wavelet-like transform, generated by the Poisson integral and a “wavelet measure.” By making use of the relevant Calderón-type reproducing formula, we obtain an explicit inversion formula for the Flett potentials which are interpreted as negative fractional powers of the operator (E+Λ), where Λ=(−Δ)^1/2, Δ is the Laplacian and E is the identity operator.     

Exceptional sets and wavelet packets orthonormal bases

We give a partial positive answer to a problem posed by Coifman et al. in [1]. Indeed, starting from the transfer function m₀ arising from the Meyer wavelet and assuming m₀=1 only on [–/3, /3], we provide an example of pairwise disjoint dyadic intervals of the form I(n, q)=[2^qn, 2^q(n+1)), (n, q)EN×Z, which cover [0, +) except for a set A of Hausdorff dimension equal to 1/2, and such that the corresponding wavelet packets 2^q/2w_n (2^qx–k), kZ, (n, q)EN×Z form an orthonormal basis of L²(R).     

Wavelet Galerkin method for eigenvalue problem of a compact integral operator

We consider the approximation of eigenfunctions of a compact integral operator with a smooth kernel by the Galerkin method using wavelet bases. By truncating the Galerkin operator, we obtain a sparse representation of a matrix eigenvalue problem. We prove that the error bounds for the eigenvalues and for the distance between the spectral subspaces are of the orders O(nμ^-2nr) and O(μ^-nr), respectively, where μ⁻ⁿ denotes the norm of the partition and r denotes the order of the wavelet basis functions. By iterating the eigenvectors, we show that the error bounds for the eigenvectors are of the order O(nμ^-2nr). We illustrate our results with numerical results.     

On Supersets of Wavelet Sets

Considering a single dyadic orthonormal wavelet ψ in L ²(?), it is still an open problem whether the support of $\widehat{\psi}$ always contains a wavelet set. As far as we know, the only result in this direction is that if the Fourier support of a wavelet function is “small” then it is either a wavelet set or a union of two wavelet sets. Without assuming that a set S is the Fourier support of a wavelet, we obtain some necessary conditions and some sufficient conditions for a “small” set S to contain a wavelet set. The main results, which are in terms of the relationship between two explicitly constructed subsets A and B of S and two subsets T ₂ and D ₂ of S intersecting itself exactly twice translationally and dilationally respectively, are (1) if $A\cup B\not\subseteq T_{2}\cap D_{2}$ then S does not contain a wavelet set; and (2) if A∪B?T ₂∩D ₂ then every wavelet subset of S must be in S?(A∪B) and if S?(A∪B) satisfies a “weak” condition then there exists a wavelet subset of S?(A∪B). In particular, if the set S?(A∪B) is of the right size then it must be a wavelet set.     

Walsh-Type Wavelet Packet Expansions

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 L^p( ), 1<p<∞, and we construct an explicit function in L¹( ) for which the expansion fails. Then we prove that expansions of L^p( )-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 L^p[0,1).     

Multiresolution analysis on local fields

In this paper, we first establish the fundamental framework of multiresolution on local field in wavelet analysis. Accurately, orthonormal systems consisting of integer translations of a single function φ∈L²(K) are characterized (Theorems 1 and 2), the concept of multiresolution analysis on local field (Definition 3) and construction of corresponding wavelet vectors (Theorem 3) are given. The definition of integral periodicity of a function on local field (Definition 4) is also given. An example is finally presented.