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.     

On <Emphasis Type="Italic">s</Emphasis>-Elementary Super Frame Wavelets and Their Path-Connectedness

A super wavelet of length n is an n-tuple (ψ ₁,ψ ₂,…,ψ _n ) in the product space \(\prod_{j=1}^{n} L^{2}(\mathbb{R})\), such that the coordinated dilates of all its coordinated translates form an orthonormal basis for \(\prod_{j=1}^{n} L^{2} (\mathbb{R})\). This concept is generalized to the so-called super frame wavelets, super tight frame wavelets and super normalized tight frame wavelets (or super Parseval frame wavelets), namely an n-tuple (η ₁,η ₂,…,η _n ) in \(\prod_{j=1}^{n}L^{2} (\mathbb{R})\) such that the coordinated dilates of all its coordinated translates form a frame, a tight frame, or a normalized tight frame for \(\prod_{j=1}^{n} L^{2}(\mathbb{R})\). In this paper, we study the super frame wavelets and the super tight frame wavelets whose Fourier transforms are defined by set theoretical functions (called s-elementary frame wavelets). An m-tuple of sets (E ₁,E ₂,…,E _m ) is said to be τ-disjoint if the E _j ’s are pair-wise disjoint under the 2π-translations. We prove that a τ-disjoint m-tuple (E ₁,E ₂,…,E _m ) of frame sets (i.e., η _j defined by \(\widehat{\eta_{j}}=\frac{1}{\sqrt{2\pi}}\chi_{E_{j}}\) is a frame wavelet for L ²(?) for each j) lead to a super frame wavelet (η ₁,η ₂,…,η _m ) for \(\prod_{j=1}^{m} L^{2} (\mathbb{R})\) where \(\widehat{\eta_{j}}=\frac{1}{\sqrt{2\pi}}\chi_{E_{j}}\). In the case of super tight frame wavelets, we prove that (η ₁,η ₂,…,η _m ), defined by \(\widehat{\eta_{j}}=\frac{1}{\sqrt{2\pi}}\chi_{E_{j}}\), is a super tight frame wavelet for ∏_1≤j≤m L ²(?) with frame bound k ₀ if and only if each η _j is a tight frame wavelet for L ²(?) with frame bound k ₀ and that (E ₁,E ₂,…,E _m ) is τ-disjoint. Denote the set of all τ-disjoint s-elementary super frame wavelets for ∏_1≤j≤m L ²(?) by \(\mathfrak{S}(m)\) and the set of all s-elementary super tight frame wavelets (with the same frame bound k ₀) for ∏_1≤j≤m L ²(?) by \(\mathfrak{S}^{k_{0}}(m)\). We further prove that \(\mathfrak{S}(m)\) and \(\mathfrak{S}^{k_{0}}(m)\) are both path-connected under the ∏_1≤j≤m L ²(?) norm, for any given positive integers m and k ₀.     

On the direct path problem of s-elementary frame wavelets

In this paper, we discuss the path-connectivity between two s-elementary normalized tight frame wavelets via the so-called direct paths. We show that the existence of such a direct path is equivalent to the non-existence of an atom of a σ-algebra defined over the defining sets of the corresponding frame wavelets, using a mapping defined by the natural translation and dilation operations between the sets. In particular, this gives an equivalent condition for the existence of a direct path between two s-elementary wavelets.     

On Parseval super-frame wavelets

Suppose that η1,...,ηn are measurable functions in L2(R).We call the n-tuple (η1,…,ηn) a Parseval super frame wavelet of length n if {2k/2η1(2kt-)（@）...(@)2k/2ηn(2kt-l):k,l∈Z}is a Parseval frame for L2...     

The characterization of a class of multivariate MRA and semi-orthogonal parseval frame wavelets

Let A be a d × d expansive matrix with ∣detA∣ = 2. This paper addresses Parseval frame wavelets (PFWs) in the setting of reducing subspaces of L²(R^d). We prove that all semi-orthogonal PFWs (semi-orthogonal MRA PFWs) are precisely the ones with their dimension functions being non-negative integer-valued (0 or 1). We also characterize all MRA PFWs. Some examples are provided.     

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.     

Tight frames of multidimensional wavelets

In this paper we deal with multidimensional wavelets arising from a multiresolution analysis with an arbitrary dilation matrix A, namely we have scaling equations $$\varphi ^s (x) = \sum\limits_{k \in \mathbb{Z}^n } {h_k^s \sqrt {|\det A|} \varphi ^1 } (Ax - k) for s = 1, \ldots ,q,$$ where ?¹ is a scaling function for this multiresolution and ?², …, ?^q (q=|det A |) are wavelets. Orthogonality conditions for ?¹, …, ?^q naturally impose constraints on the scaling coefficients $\{ h_k^s \} _{k \in \mathbb{Z}^n }^{s = 1, \ldots ,q} $ , which are then called the wavelet matrix. We show how to reconstruct functions satisfying the scaling equations above and show that ?², …, ?^q always constitute a tight frame with constant 1. Furthermore, we generalize the sufficient and necessary conditions of orthogonality given by Lawton and Cohen to the case of several dimensions and arbitrary dilation matrix A.     

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.     

Frame Wavelets with Compact Supports for L^2（R^n）

The construction of frame wavelets with compact supports is a meaningful problem in wavelet analysis. In particular, it is a hard work to construct the frame wavelets with explicit analytic forms. For a given n × n real expansive matrix A, the frame-sets with respect to A are a family of sets in R^n. Based on the frame-sets, a class of high-dimensional frame wavelets with analytic forms are constructed, which can be non-bandlimited, or even compactly supported. As an application, the construction is illustrated by several examples, in which some new frame wavelets with compact supports are constructed. Moreover, since the main result of this paper is about general dilation matrices, in the examples we present a family of frame wavelets associated with some non-integer dilation matrices that is meaningful in computational geometry.     

Characterization and Connectivity of Generalized Filters in L 2(ℝ d )

Let A be a d×d real expansive integer matrix (i.e., a matrix with real entries whose eigenvalues are all of modules greater than one) with |det?A|=2, and let m (which is called A-dilation generalized filter) be a 2π? ^d periodic function with the property that |m(s)|²+|m(s+2π h ₂)|²=1, where h ₂∈(A ^τ )^?1? ^d ?? ^d . In this paper, we characterize the set of all A-dilation generalized filters and show that this set is path-connected in $L^{2}({\mathbb{T}}^{d})$ -norm by using the technique of filter multipliers. We also obtain an equivalent condition for an A-dilation generalized filter to be an A-dilation low pass filter. These extend the results of Manos Papadakis et al. from one dimensional case to high dimensions and matrix dilations cases.     

A-Parseval框架小波的特征刻画

研究与伸缩矩阵A相关的Parseval框架小波(A-PFW)的特征刻画,其中伸缩矩阵A满足A~3=2I_3且A的每一列元素之和均为偶数.首先,讨论了与两个特殊伸缩矩阵B,C相关的Parseval框架小波(B-PFW,C-PFW)之间的关系,并得到C-PFW分别与两类特殊伸缩矩阵D,■相关的Parseval框架小波(D-PFW,■-PFW)之间的等价关系.其次,探讨了伪尺度函数和源于多分辨分析的A-PFW(MRA A-PFW)的特征刻画.最后,借助于维数函数,给出了A-PFW是MRA A-PFW的一个充要条件.     

Multipliers, Phases and Connectivity of MRA Wavelets in L 2(ℝ2)

Let A be any 2×2 real expansive matrix. For any A-dilation wavelet ψ, let [^(y)]\widehat{\psi} be its Fourier transform. A measurable function f is called an A-dilation wavelet multiplier if the inverse Fourier transform of (f[^(y)])(f\widehat{\psi}) is an A-dilation wavelet for any A-dilation wavelet ψ. In this paper, we give a complete characterization of all A-dilation wavelet multipliers under the condition that A is a 2×2 matrix with integer entries and |{det }(A)|=2. Using this result, we are able to characterize the phases of A-dilation wavelets and prove that the set of all A-dilation MRA wavelets is path-connected under the L ²(ℝ²) norm topology for any such matrix A.     

The s-elementary frame wavelets are path connected

An s-elementary frame wavelet is a function which is a frame wavelet and is defined by a Lebesgue measurable set such that . In this paper we prove that the family of s-elementary frame wavelets is a path-connected set in the -norm. This result also holds for s-elementary -dilation frame wavelets in in general. On the other hand, we prove that the path-connectedness of s-elementary frame wavelets cannot be strengthened to uniform path-connectedness. In fact, the sets of normalized tight frame wavelets and frame wavelets are not uniformly path-connected either.

     

The Equivalence Between Seven Classes of Wavelet Multipliers and Arcwise Connectivity They Induce

Let A be a d×d expansive matrix with |detA|=2. An A-wavelet is a function $\psi\in L^{2}(\mathbb{R}^{d})$ such that $\{2^{\frac{j}{2}}\psi(A\cdot-k):\,j\in \mathbb{Z},\,k\in \mathbb{Z}^{d}\}$ is an orthonormal basis for $L^{2}(\mathbb{R}^{d})$ . A measurable function f is called an A-wavelet multiplier if the inverse Fourier transform of $f\hat{\psi}$ is an A-wavelet whenever ψ is an A-wavelet, where $\hat{\psi}$ denotes the Fourier transform of ψ. A-scaling function multiplier, A-PFW multiplier, semi-orthogonal A-PFW multiplier, MRA A-wavelet multiplier, MRA A-PFW multiplier and semi-orthogonal MRA A-PFW multiplier are defined similarly. In this paper, we prove that the above seven classes of multipliers are equivalent, and obtain a characterization of them. We then prove that if the set of all A-wavelet multipliers acts on some A-scaling function (A-wavelet, A-PFW, semi-orthogonal A-PFW, MRA A-wavelet, MRA A-PFW, semi-orthogonal MRA A-PFW), the orbit is arcwise connected in $L^{2}(\mathbb{R}^{d})$ , and that if the generator of an orbit is an MRA A-PFW, the orbit is equal to the set of all MRA A-PFWs whose Fourier transforms have same module, and is also equal to the set of all MRA A-PFWs with corresponding pseudo-scaling functions having the same module of their Fourier transforms.     

The Closure of the Set of Tight Frame Wavelets

We study properties of the closure of the set of tight frame wavelets. We give a necessary condition and a sufficient condition for a function to be in this closure. In particular, we show that the collection of tight frame wavelets is not dense in L ²(? ⁿ ), which answers a question posed by D. Han and D. Larson (Preprint, 2008).     

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.      

Units of compatible nearrings

We study singly-generated wavelet systems on ${\mathbb {R}^2}$ that are naturally associated with rank-one wavelet systems on the Heisenberg group N. We prove a necessary condition on the generator in order that any such system be a Parseval frame. Given a suitable subset I of the dual of N, we give an explicit construction for Parseval frame wavelets that are associated with I. We say that ${g\in L^2(I\times \mathbb {R})}$ is Gabor field over I if, for a.e. ${\lambda \in I}$ , |??|^1/2 g(??, ·) is the Gabor generator of a Parseval frame for ${L^2(\mathbb {R})}$ , and that I is a Heisenberg wavelet set if every Gabor field over I is a Parseval frame (mother-)wavelet for ${L^2(\mathbb {R}^2)}$ . We then show that I is a Heisenberg wavelet set if and only if I is both translation congruent with a subset of the unit interval and dilation congruent with the Shannon set.     

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.     

Subdivision schemes with polynomially decaying masks

In this paper we investigate the L ₂-solutions of vector refinement equations with polynomially decaying masks and a general dilation matrix, which plays a vital role for characterizations of wavelets and biorthogonal wavelets with infinite support. A vector refinement equation with polynomially decaying masks and a general dilation matrix is the form:

$ \phi(x)=\sum_{\alpha\in\Bbb Z^s}a(\alpha)\medspace\phi(Mx-\alpha),\quad x\in\Bbb R^s, $

where the vector of functions \(\phi=(\phi_{1},\cdots,\phi_{r})^{T}\) is in \((L_{2}(\Bbb R^s))^{r},\) \(a:=(a(\alpha))_{\alpha\in\Bbb Z^s}\) is a polynomially decaying sequence of r×r matrices called refinement mask and M is an s×s integer matrix such that \(\lim_{n\to\infty}M^{-n}=0.\) The corresponding cascade operator on \((L_2(\Bbb R^s))^r\) is given by:

$ Q_{a}f(x):=\sum_{\alpha\in\Bbb Z^s}a(\alpha)f(Mx-\alpha),\quad x\in\Bbb R^s, \quad f=(f_1,...,f_r)^T\in (L_2(\Bbb R^s))^r. $

The iterative scheme \((Q_a^nf)_{n=1,2,\cdots,}\) is called vector cascade algorithm. In this paper we give a complete characterization of convergence of the sequence \((Q_a^nf)_{n=1,2\cdots}\) in L ₂-norm. Some properties of the transition operator restricted to a certain linear space are discussed. As an application of convergence, we also obtain a characterization of smoothness of solutions of refinement equation mentioned above for the case r?=?1.