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.     

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 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.     

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.     

Construction and decomposition of biorthogonal vector-valued wavelets with compact support

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.     

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).     

矩阵伸缩的多元多重向量值小波包的双正交性

1引言 小波分析是二十世纪八十年代中期发展起来的一个数学分枝,其应用涉及自然科学与工程技术的许多领域[1-3].向量值小波从属多小波理论范畴.文献[4]引入向量值小波的概念,讨论了多重向量值双正交小波的存在性及其构造.Bacchelli等[5]证明了多重向量值双正交小波的存在性.文献[6]运用多重向量值双正交小波变换研究海洋涡流现象.     

Wavelet characterization of weighted spaces

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.     

Lipschitz continuity and Gateaux differentiability of the best approximation operator in vector-valued Chebyshev approximation

When G is a finite-dimensional Haar subspace of C(X,R^k), the vector-valued functions (including complex-valued functions when k is 2) from a finite set X to Euclidean k-dimensional space, it is well-known that at any function f in C(X,R^k) the best approximation operator satisfies the strong unicity condition of order 2 and a Lipschitz (Hőlder) condition of order . This note shows that in fact the best approximation operator satisfies the usual Lipschitz condition of order 1 and has a Gateaux derivative on a dense set of functions in C(X,R^k).     

Tests for the mean direction of the Langevin distribution with large concentration parameter

In this paper we study the asymptotic behaviors of the likelihood ratio criterion (T_L^(s)), Watson statistic (T_W^(s)) and Rao statistic (T_R^(s)) for testing H_0s: μ (a given subspace) against H_1s: μ , based on a sample of size n from a p-variate Langevin distribution M_p(μ, κ) when κ is large. For the case when κ is known, asymptotic expansions of the null and nonnull distributions of these statistics are obtained. It is shown that the powers of these statistics are coincident up to the order κ⁻¹. For the case when κ is unknown, it is shown that T_R^(s) T_L^(s) T_W^(s) in their powers up to the order κ⁻¹.     

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.     

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.     

Completeness of Integer Translates in Function Spaces on

We show that each of the Banach spacesC₀( ) andL^p( ), 2<p<∞, contains a function whose integer translates are complete. This function can also be chosen so that one of the following additional conditions hold: (1) Its non-negative integer translates are already complete. (2) Its integer translates form an orthonormal system inL²( ). (3) Its integer translates form a minimal system. A similar result holds for the corresponding Sobolev space, for certain weightedL²spaces, and in the multivariate setting. We also prove some results in the opposite direction.     

Lipschitz continuity of the best approximation operator in vector-valued Chebyshev approximation

When G is a finite dimensional Haar subspace of C(X,R^k), the vector-valued continuous functions (including complex-valued functions when k is 2) from a finite set X to Euclidean k-dimensional space, it is well-known that at any function f in C(X,R^k) the best approximation operator satisfies the strong unicity condition of order 2 and a Lipschitz (Hőlder) condition of order . This note shows that in fact the best approximation operator satisfies the usual Lipschitz condition of order 1.     

The Norm and Regularized Trace of the Cauchy Transform

In this paper, the norm of the Cauchy transform C is obtained on the space L ²(D, dμ), where dμ = ω(|z|) dA(z). Also, (for the case ω ≡ 1), the first regularized trace of the operator C* C on L ²(Ω) is obtained. The results are illustrated by examples, with different specific choices of the function ω and the domain Ω.__________Translated from Matematicheskie Zametki, vol. 77, no. 6, 2005, pp. 844–853.Original Russian Text Copyright ©2005 by M. R. Dostanic.     

On wavelet decomposition of Hermite type splines

We consider wavelet decompositions of spaces of Hermite type splines of class C¹(α, β) that are defined by a 4-component vector-valued function ϕ(t) ∈ C¹ (α, β) by means of a grid X (not necessarily uniform) on (α, β) ∈ ℝ¹ (the special case ϕ(t)^def = (1, t, t²,t³)^T corresponds to cubic Hermite splines). The basis wavelets obtained are compactly supported. The decomposition and reconstruction formulas are given. Bibliography: 8 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 35, 2007, pp. 33–46     

Wavelets on the 2-Sphere: A Group-Theoretical Approach

We present a purely group-theoretical derivation of the continuous wavelet transform (CWT) on the 2-sphere S², based on the construction of general coherent states associated to square integrable group representations. The parameter space X of our CWT is the product of SO(3) for motions and ⁺_* for dilations on S², which are embedded into the Lorentz group SO₀(3, 1) via the Iwasawa decomposition, so that X SO₀(3, 1) M Y S O L N, where N . We select an appropriate unitary representation of SO₀(3, 1) acting in the space L²(S², d μ) of finite energy signals on S². This representation is square integrable over X; thus it yields immediately the wavelets on S² and the associated CWT. We find a necessary condition for the admissibility of a wavelet, in the form of a zero mean condition. Finally, the Euclidean limit of this CWT on S² is obtained by redoing the construction on a sphere of radius R and performing a group contraction for R → ∞. Then the parameter space goes into the similitude group of ² and one recovers exactly the CWT on the plane, including the usual zero mean necessary condition for admissibility.     

BOUNDEDNESS OF VECTOR-VALUED CALDERON-ZYGMUND OPERATORS ON HERZ SPACES WITH NON-DOUBLING MEASURES

In the paper we obtain vector-valued inequalities for Calderon-Zygmund operator,simply CZO On Herz space and weak Herz space.In particular,we obtain vector-valued inequalities for CZO on Lq(Rd,│x│αdμ)space,with 1＜q＜∞,-n＜α＜n(q-1),and on L1,∞(Rd,│x│αdμ)space,with -n＜α＜0.     

Spherical Harmonic Analysis for Spinors on H n (C)

Recent results on the harmonic analysis of spinor fields on the complex hyperbolic space H ⁿ (C) are reviewed. We discuss the action of the invariant differential operators on the Poisson transforms, the theory of spherical functions and the spherical transform. The inversion formula, the Paley–Wiener theorem, and the Plancherel theorem for the spherical transform are obtained by reduction to Jacobi analysis on L ²(R).     

On semi-fredholm properties of a boundary value problem inR + n

The paper considers a boundary value problem with the help of the smallest closed extensionL ^∼ :H _k →H _{k 0}×B _{h 1}×...×B _{h N} of a linear operatorL :C ₍₀₎ ^∞ (R ₊ ⁿ ) →L(R ₊ ⁿ )×L(R ⁿ⁻¹)×...×L(R ⁿ⁻¹). Here the spacesH _k (the spaces ℬ _h ) are appropriate subspaces ofD′(R ₊ ⁿ ) (ofD′(R ⁿ⁻¹), resp.),L(R ₊ ⁿ ) andC ₍₀₎ ^∞ (R ₊ ⁿ )) denotes the linear space of smooth functionsR ⁿ →C, which are restrictions onR ₊ ⁿ of a function from the Schwartz classL (fromC ₀ ^∞ , resp.),L(R ⁿ⁻¹) is the Schwartz class of functionsR ⁿ⁻¹ →C andL is constructed by pseudo-differential operators. Criteria for the closedness of the rangeR(L ^∼) and for the uniqueness of solutionsL ^∼ U=F are expressed. In addition, ana priori estimate for the corresponding boundary value problem is established.