Some simple Haar-type wavelets in higher dimensions

An orthonormal wavelet system in ℝ^d, d ∈ ℕ, is a countable collection of functions {ψ _j,k ^ℓ }, j ∈ ℤ, k ∈ ℤ^d, ℓ = 1,..., L, of the form that is an orthonormal basis for L² (ℝ^d), where a ∈ GL_d (ℝ) is an expanding matrix. The first such system to be discovered (almost 100 years ago) is the Haar system for which L = d = 1, ψ¹(x) = ψ(x) = κ[0,1/2)^(x) − κ_[l/2,1) ^(x), a = 2. It is a natural problem to extend these systems to higher dimensions. A simple solution is found by taking appropriate products Φ(x₁, x₂, ..., x_d) = φ₁ (x₁)φ₂(x₂) ... φ_d(x_d) of functions of one variable. The obtained wavelet system is not always convenient for applications. It is desirable to find “nonseparable” examples. One encounters certain difficulties, however, when one tries to construct such MRA wavelet systems. For example, if a = ( _1-1 ^{1 1} ) is the quincunx dilation matrix, it is well-known (see, e.g., [5]) that one can construct nonseparable Haar-type scaling functions which are characteristic functions of rather complicated fractal-like compact sets. In this work we shall construct considerably simpler Haar-type wavelets if we use the ideas arising from “composite dilation” wavelets. These were developed in [7] and involve dilations by matrices that are products of the form a^jb, j ∈ ℤ, where a ∈ GL_d(ℝ) has some “expanding” property and b belongs to a group of matrices in GL_d(ℝ) having |det b| = 1.     

Functional Gabor frame multipliers

A Gabor frame multiplier is a bounded operator that maps normalized tight Gabor frame generators to normalized tight Gabor frame generators. While characterization of such operators is still unknown, we give a complete characterization for the functional Gabor frame multipliers. We prove that a L^∞ -function h is a functional Gabor frame multiplier (for the time-frequency lattice aℤ × bℤ) if and only if it is unimodular and is a-periodic. Along the same line, we also characterize all the Gabor frame generators g (resp. frame wavelets ψ) for which there is a function ∈ L^∞(ℝ) such that {wg_mn} (resp. ωψ_k,ℝ) is a normalized tight frame.     

Subspace Weyl-Heisenberg frames

A Weyl-Heisenberg frame (WH frame) for L²(ℝ) allows every square integrable function on the line to be decomposed into the infinite sum of linear combination of translated and modulated versions of a fixed function. Some sufficient conditions for g ∈ L²(ℝ) to be a subspace Weyl-Heisenberg frame were given in a recent work [3] by Casazza and Christensen. Obviously every invariant subspace (under translation and modulation) is cyclic if it has a subspace WH frame. In the present article we prove that the cyclicity property is also sufficient for a subspace to admit a WH frame. We also investigate the dilation property for subspace Weyl-Heisenberg frames and show that every normalized tight subspace WH frame can be dilated to a normalized tight WH frame which is “maximal” In other words, every subspace WH frame is the compression of a WH frame which cannot be dilated anymore within the L²(ℝ) space. Communicated by Hans G. Feichtinger     

Large classes of minimally supported frequency wavelets of<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>(ℝ) and<Emphasis Type="Italic">H</Emphasis><Superscript>2</Superscript>(ℝ)

We introduce a new method to construct large classes of minimally supported frequency (MSF) wavelets of the Hardy space H ² (ℝ)and symmetric MSF wavelets of L ² (ℝ),and discuss the classification of such wavelets. As an application, we show that there are uncountably many such wavelet sets of L ² (ℝ)and H ² (ℝ).We also enumerate some of the symmetric wavelet sets of L ² (ℝ)and all wavelet sets of H ² (ℝ)consisting of three intervals. Finally, we construct families of MSF wavelets of L ₂ (ℝ)with Fourier transform even and not vanishing in any neighborhood of the origin.     

The Calderón reproducing formula, windowed X-ray transforms, and radon transforms in LP-spaces

The generalized Calderón reproducing formula involving “wavelet measure” is established for functions f ∈ L^p(ℝⁿ). The special choice of the wavelet measure in the reproducing formula gives rise to the continuous decomposition of f into wavelets, and enables one to obtain inversion formulae for generalized windowed X-ray transforms, the Radon transform, and k-plane transforms. The admissibility conditions for the wavelet measure μ are presented in terms of μ itself and in terms of the Fourier transform of μ. Acknowledgements and Notes. Partially sponsored by the Edmund Landau Center for research in Mathematical Analysis, supported by the Minerva Foundation (Germany).     

A characterization of the higher dimensional groups associated with continuous wavelets

A subgroup D of GL (n, ℝ) is said to be admissible if the semidirect product of D and ℝ ⁿ , considered as a subgroup of the affine group on ℝ ⁿ , admits wavelets ψ ∈ L²(ℝ ⁿ ) satisfying a generalization of the Calderón reproducing, formula. This article provides a nearly complete characterization of the admissible subgroups D. More precisely, if D is admissible, then the stability subgroup D_x for the transpose action of D on ℝ ⁿ must be compact for a. e. x. ∈ ℝ ⁿ ; moreover, if Δ is the modular function of D, there must exist an a ∈ D such that |det a| ≠ Δ(a). Conversely, if the last condition holds and for a. e. x ∈ ℝ ⁿ there exists an ε > 0 for which the ε-stabilizer D _x ^ε is compact, then D is admissible. Numerous examples are given of both admissible and non-admissible groups.     

A unified characterization of reproducing systems generated by a finite family,II

By a “reproducing” method forH =L ²(ℝ ⁿ ) we mean the use of two countable families {e _α : α ∈A}, {f _α : α ∈A}, inH, so that the first “analyzes” a function h ∈H by forming the inner products {<h,e _α >: α ∈A} and the second “reconstructs” h from this information:h = Σα∈A <h,e _α >:f _α. A variety of such systems have been used successfully in both pure and applied mathematics. They have the following feature in common: they are generated by a single or a finite collection of functions by applying to the generators two countable families of operators that consist of two of the following three actions: dilations, modulations, and translations. The Gabor systems, for example, involve a countable collection of modulations and translations; the affine systems (that produce a variety of wavelets) involve translations and dilations. A considerable amount of research has been conducted in order to characterize those generators of such systems. In this article we establish a result that “unifies” all of these characterizations by means of a relatively simple system of equalities. Such unification has been presented in a work by one of the authors. One of the novelties here is the use of a different approach that provides us with a considerably more general class of such reproducing systems; for example, in the affine case, we need not to restrict the dilation matrices to ones that preserve the integer lattice and are expanding on ℝ ⁿ . Another novelty is a detailed analysis, in the case of affine and quasi-affine systems, of the characterizing equations for different kinds of dilation matrices.     

Orthogonality criteria for compactly supported refinable functions and refinable function vectors

A refinable function φ(x):ℝⁿ→ℝ or, more generally, a refinable function vector Φ(x)=[φ₁(x),...,φ_r(x)]^T is an L¹ solution of a system of (vector-valued) refinement equations involving expansion by a dilation matrix A, which is an expanding integer matrix. A refinable function vector is called orthogonal if {φ_j(x−α):α∈ℤⁿ, 1≤j≤r form an orthogonal set of functions in L²(ℝⁿ). Compactly supported orthogonal refinable functions and function vectors can be used to construct orthonormal wavelet and multiwavelet bases of L²(ℝⁿ). In this paper we give a comprehensive set of necessary and sufficient conditions for the orthogonality of compactly supported refinable functions and refinable function vectors.     

On wavelet sets

It is proved that associated with every wavelet set is a closely related “regularized” wavelet set which has very nice properties. Then it is shown that for many (and perhaps all) pairs E, F, of wavelet sets, the corresponding MSF wavelets can be connected by a continuous path in L²(ℝ) of MSF wavelets for which the Fourier transform has support contained in E ∪ F. Our technique applies, in particular, to the Shannon and Journe wavelet sets.     

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.     

Dual Wavelet Frames and Riesz Bases in Sobolev Spaces

This paper generalizes the mixed extension principle in L ₂(ℝ ^d ) of (Ron and Shen in J. Fourier Anal. Appl. 3:617–637, 1997) to a pair of dual Sobolev spaces H ^s (ℝ ^d ) and H ^−s (ℝ ^d ). In terms of masks for φ,ψ ¹,…,ψ ^L ∈H ^s (ℝ ^d ) and , simple sufficient conditions are given to ensure that (X ^s (φ;ψ ¹,…,ψ ^L ), forms a pair of dual wavelet frames in (H ^s (ℝ ^d ),H ^−s (ℝ ^d )), where

Smoothing minimally supported frequency wavelets: Part II

The main purpose of this paper is to give a procedure to “mollify” the low-pass filters of a large number ofMinimally Supported Frequency (MSF) wavelets so that the smoother functions obtained in this way are also low-pass filters for an MRA. Hence, we are able to approximate (in the L ² -norm) MSF wavelets by wavelets with any desired degree of smoothness on the Fourier transform side. Although the MSF wavelets we consider are bandlimited, this may not be true for their smooth approximations. This phenomena is related to the invariant cycles under the transformation x ↦2x (mod2π). We also give a characterization of all low-pass filters for MSF wavelets. Throughout the paper new and interesting examples of wavelets are described.     

From continuous to discrete Weyl-Heisenberg frames through sampling

In this article we consider the question when one can generate a Weyl- Heisenberg frame for l ² (ℤ) with shift parameters N, M ⁻¹ (integer N, M) by sampling a Weyl-Heisenberg frame for L ² (ℝ) with the same shift parameters at the integers. It is shown that this is possible when the window g ε L ² (ℝ) generating the Weyl-Heisenberg frame satisfies an appropriate regularity condition at the integers. When, in addition, the Tolimieri-Orr condition A is satisfied, the minimum energy dual window ^o γ ε L ² (ℝ) can be sampled as well, and the two sampled windows continue to be related by duality and minimality. The results of this article also provide a rigorous basis for the engineering practice of computing dual functions by writing the Wexler-Raz biorthogonality condition in the time-domain as a collection of decoupled linear systems involving samples of g and ^o γ as knowns and unknowns, respectively. We briefly indicate when and how one can generate a Weyl-Heisenberg frame for the space of K-periodic sequences, where K=LCM (N, M), by periodization of a Weyl-Heisenberg frame for ℓ ² ℤ with shift parameters N, M ⁻¹ .     

Construction of multivariate compactly supported orthonormal wavelets

We propose a constructive method to find compactly supported orthonormal wavelets for any given compactly supported scaling function φ in the multivariate setting. For simplicity, we start with a standard dilation matrix 2I_2×2 in the bivariate setting and show how to construct compactly supported functions ψ₁,. . .,ψ_n with n>3 such that {2^kψ_j(2^kx−ℓ,2^ky−m), k,ℓ,m∈Z, j=1,. . .,n} is an orthonormal basis for L₂(ℝ²). Here, n is dependent on the size of the support of φ. With parallel processes in modern computer, it is possible to use these orthonormal wavelets for applications. Furthermore, the constructive method can be extended to construct compactly supported multi-wavelets for any given compactly supported orthonormal multi-scaling vector. Finally, we mention that the constructions can be generalized to the multivariate setting. Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday Mathematics subject classifications (2000) 42C15, 42C30.     

Affine Synthesis onto L p when 0<p≤1

The affine synthesis operator is shown to map the coefficient space ℓ ^p (ℤ₊×ℤ ^d ) surjectively onto L ^p (ℝ ^d ), for p∈(0,1]. Here ψ _j,k (x)=|det a _j |^1/p ψ(a _j x−k) for dilation matrices a _j that expand, and the synthesizer ψ∈L ^p (ℝ ^d ) need satisfy only mild restrictions, for example, ψ∈L ¹(ℝ ^d ) with nonzero integral or else with periodization that is real-valued, nontrivial and bounded below. An affine atomic decomposition of L ^p follows immediately:

A characterization of wavelet families arising from biorthogonal MRA’s of multiplicityd

In this article we give a necessary and sufficient condition for a pair of wavelet families in L²(ℝ ⁿ ), to arise from a pair of biorthogonal MRA’s. The condition is given in terms of simple equations involving the functions ψ^ℓ and . To work in greater generality, we allow multiresolution analyses of arbitrary multiplicity, based on lattice translations and matrix dilations. Our result extends the characterization theorem of G. Gripenberg and X. Wang for dyadic orthonormal wavelets in L²(ℝ),and includes, as particular cases, the sufficient conditions of P. Auscher and P.G. Lemarié in the biorthogonal situation.     

A unified characterization of reproducing systems generated by a finite family

This article presents a general result from the study of shift-invariant spaces that characterizes tight frame and dual frame generators for shift-invariant subspaces of L²(ℝⁿ). A number of applications of this general result are then obtained, among which are the characterization of tight frames and dual frames for Gabor and wavelet systems.     

Oscillating multipliers for some eigenfunction expansions

Let P be a non-negative, self-adjoint differential operator of degree d on ℝⁿ. Assume that the associated Bochner-Riesz kernel s _R ^δ satisfies the estimate, |s _R ^δ (x, y)| ≤ C R^n/d(1+R^1/d|x - y|^-αδ+β)for some fixed constants a>0 and β. We study L^p boundedness of operators of the form m(P), m coming from the symbol class S _p ^−α . We prove that m(P) is bounded on L^P if . We also study multipliers associated to the Hermite operator H on ℝⁿ and the special Hermite operator L on ℂⁿ given by the symbols . As a special case we obtain L^p boundedness of solutions to the Wave equation associated to H and L.     

Irregular wavelet frames on<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>(ℝ<Superscript>n</Superscript>)

In this paper, we present the conditions on dilation parameter {s _j}_j that ensure a discrete irregular wavelet system {s _j ^n/2ψ(s _j ·−bk)} _{j∈ℤ,k∈ℤ} ⁿ to be a frame on L²(ℝⁿ), and for the wavelet frame we consider the perturbations of translation parameter b and frame function ψ respectively.     

Dilations and Completions for Gabor Systems

Let be a full rank time-frequency lattice in ℝ ^d ×ℝ ^d . In this note we first prove that any dual Gabor frame pair for a Λ-shift invariant subspace M can be dilated to a dual Gabor frame pair for the whole space L ²(ℝ ^d ) when the volume v(Λ) of the lattice Λ satisfies the condition v(Λ)≤1, and to a dual Gabor Riesz basis pair for a Λ-shift invariant subspace containing M when v(Λ)>1. This generalizes the dilation result in Gabardo and Han (J. Fourier Anal. Appl. 7:419–433, [2001]) to both higher dimensions and dual subspace Gabor frame pairs. Secondly, for any fixed positive integer N, we investigate the problem whether any Bessel–Gabor family G(g,Λ) can be completed to a tight Gabor (multi-)frame G(g,Λ)∪(∪ _j=1 ^N G(g _j ,Λ)) for L ²(ℝ ^d ). We show that this is true whenever v(Λ)≤N. In particular, when v(Λ)≤1, any Bessel–Gabor system is a subset of a tight Gabor frame G(g,Λ)∪G(h,Λ) for L ²(ℝ ^d ). Related results for affine systems are also discussed. Communicated by Chris Heil.