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.     

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.     

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.     

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.     

The construction of single wavelets in d-dimensions

Sets K in d-dimensional Euclidean space are constructed with the property that the inverse Fourier transform of the characteristic function 1 _K is a single dyadic orthonormal wavelet. The construction is characterized by its generality in the procedure, by its computational implementation, and by its simplicity. The general case in which the inverse Fourier transforms of the characteristic functions 1K ¹, ..., 1K ^L are a family of orthonormal wavelets is treated in [27].     

Monotonicity inequalities for ther-area and a degeneracy theorem forr-minimal graphs

We establish monotonicity inequalities for the r-area of a complete oriented properly immersed r-minimal hypersurface in Euclidean space under appropriate quasi-positivity assumptions on certain invariants of the immersion. The proofs are based on the corresponding first variational formula. As an application, we derive a degeneracy theorem for an entire r-minimal graph whose defining function ƒ has first and second derivatives decaying fast enough at infinity: Its Hessian operator D² ƒ has at least n − r null eigenvalues everywhere.     

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.     

Riesz sequences of translates and generalized duals with support on [0, 1]

If the integer translates of a function ø with compact support generate a frame for a subspace W of L ²(?),then it is automatically a Riesz basis for W, and there exists a unique dual Riesz basis belonging to W. Considerable freedom can be obtained by considering oblique duals, i.e., duals not necessarily belonging to W. Extending work by Ben-Artzi and Ron, we characterize the existence of oblique duals generated by a function with support on an interval of length one. If such a generator exists, we show that it can be chosen with desired smoothness. Regardless whether ø is polynomial or not, the same condition implies that a polynomial dual supported on an interval of length one exists.     

A characterization of wavelets on general lattices

In the context of a general lattice Γ in Rⁿ and a strictly expanding map M which preserves the lattice, we characterize all the wavelet families. This result generalizes the characterization of Frazier, Garrigós, Wang, and Weis about the wavelet families with Γ = Zⁿ and M = 21. In the second part of the paper, we characterize all the MSF wavelets. Moreover, we give a constructive method for the support of the Fourier transform of an MSF wavelet and apply this method by giving examples with particular attention to the quincunx lattice.     

Density of weighted wavelet frames

If ψ ∈ L²(R), Λ is a discrete subset of the affine groupA =R ⁺ ×R, and w: Λ →R ⁺ is a weight function, then the weighted wavelet system generated by ψ, Λ, and w is . In this article we define lower and upper weighted densities D _w ⁻ (Λ) and D _w ⁺ (Λ) of Λ with respect to the geometry of the affine group, and prove that there exist necessary conditions on a weighted wavelet system in order that it possesses frame bounds. Specifically, we prove that if W(ψ, Λ, w) possesses an upper frame bound, then the upper weighted density is finite. Furthermore, for the unweighted case w = 1, we prove that if W(ψ, Λ, 1) possesses a lower frame bound and D _w ⁺ (Λ⁻¹) < ∞, then the lower density is strictly positive. We apply these results to oversampled affine systems (which include the classical affine and the quasi-affine systems as special cases), to co-affine wavelet systems, and to systems consisting only of dilations, obtaining some new results relating density to the frame properties of these systems.     

Characterizations of orthonormal scale functions: A probabilistic approach

The construction of a multiresolution analysis starts with the specification of a scale function. The Fourier transform of this function is defined by an infinite product. The convergence of this product is usually discussed in the context of L ²(R).Here, we treat the convergence problem by viewing the partial products as probabilities, converging weakly to a probability defined on an appropriate sequence space. We obtain a sufficient condition for this convergence, which is also necessary in the case where the scale function is continuous. These results extend and clarify those of Cohen [2] and Hernández et al. [4]. The method also applies to more general dilation schemes that commute with translations by Z ^d .     

Construction of wavelet sets with certain self-similarity properties

A measurable set Q ⊂ R ⁿ is a wavelet set for an expansive matrix A if F ⁻¹ _(ΧQ) is an A-dilation wavelet. Dai, Larson, and Speegle [7] discovered the existence of wavelet sets in R _n associated with any real n ×n expansive matrix. In this work, we construct a class of compact wavelet sets which do not contain the origin and which are, up to a certain linear transformation, finite unions of integer translates of an integral selfaffine tile associated with the matrix B = A ^t. Some of these wavelet sets may have good potential for applications because of their tractable geometric shapes.     

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.     

Double bubbles in<Emphasis Type="Bold">S</Emphasis><Superscript>3</Superscript> and<Emphasis Type="Bold">H</Emphasis><Superscript>3</Superscript>

We prove the double bubble conjecture in the three-sphereS ³ and hyperbolic three-spaceH ³ in the cases where we can apply Hutchings theory:

Hardy spaces and the<Emphasis Type="Italic">Tb</Emphasis> theorem

It is well-known that Calderón-Zygmund operators T are bounded on H^p for\(\frac{n}{{n + 1}}< p \leqslant 1\) provided T*(1) = 0. In this article, it is shown that if T*(b) = 0, where b is a para-accretive function, T is bounded from the classical Hardy space H^p to a new Hardy space H _b ^p . To develop an H _b ^p theory, a discrete Calderón-type reproducing formula and Plancherel-Pôlya-type inequalities associated to a para-accretive function are established. Moreover, David, Journé, and Semmes’ result [9] about the L^P, 1 < p < ∞, boundedness of the Littlewood-Paley g function associated to a para-accretive function is generalized to the case of p ≤ 1. A new characterization of the classical Hardy spaces by using more general cancellation adapted to para-accretive functions is also given. These results complement the celebrated Calderón-Zygmund operator theory.     

A characterization of semiorthogonal Parseval wavelets in abstract Hilbert spaces

A Parseval (multi)wavelet in L² (ℝ) is characterized by two requirements of its Fourier transform; the characterization of a semiorthogonal Parseval wavelet requires an additional condition of the wavelet dimension function. In this article, we use the theory of generalized multiresolution analyses to extend this idea to the more general setting of an abstract Hilbert space. We find an equation that is the abstract analog of the three conditions in L²(ℝ). Fort Lewis College     

The local part and the strong type for operators related to the Gaussian measure

In this work we present several theorems which imply the weak type 1 with respect to the Gaussian measure for the so-called local part of certain operators associated with the Ornstein-Uhlenbeck semigroup. Particular cases of these operators are the Riesz transforms of any order and the Littlewood-Paley square function. Also, we study general results based on the “size” of the operator which ensure the strong type 1 <p < ∞of both the local and global parts.     

A boundary Morera theorem

LetD ⊂C ^N ,N ≥ 2 be a bounded open set withC ² boundary and letL be an open connected set of affine complex hyperplanes inC ^N containing a hyperplane that misses . LetE = ∪_Λ∈LΛ, Γ =E ∩bD. Suppose thatf ∈C(Γ) and assume that

Holomorphic extension associated with Fourier-Legendre expansions

In this article we prove that if the coefficients of a Fourier-Legendre expansion satisfy a suitable Hausdorff-type condition, then the series converges to a function which admits a holomorphic extension to a cut-plane. Furthermore, we prove that a Laplace-type (Laplace composed with Radon) transform of the function describing the jump across the cut is the unique Carlsonian interpolation of the Fourier coefficients of the expansion. We can thus reconstruct the discontinuity function from the coefficients of the Fourier-Legendre series by the use of the Pollaczek polynomials.     

Extrapolation of Weighted norm inequalities for multivariable operators and applications

Two versions of Rubio de Francia’s extrapolation theorem for multivariable operators of functions are obtained. One version assumes an initial estimate with different weights in each space and implies boundedness on all products of Lebesgue spaces. Another version assumes an initial estimate with the same weight but yields boundedness on a product of Lebesgue spaces whose indices lie on a line. Applications are given in the context of multilinear Calderón-Zygmund operators. Vector-valued inequalities are automatically obtained for them without developing a multilinear Banach-valued theory. A multilinear extension of the Marcinkiewicz and Zygmund theorem on ℓ²-valued extensions of bounded linear operators is also obtained.