Locally Supported,Piecewise Polynomial Biorthogonal Wavelets on Nonuniform Meshes

In this paper, biorthogonal wavelets are constructed on nonuniform meshes. Both primal and dual wavelets are locally supported, continuous piecewise polynomials. The wavelets generate Riesz bases for the Sobolev spaces (H^s) for (|s| < 3/2). The wavelets at the primal side span standard Lagrange finite element spaces.     

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.     

Parseval frame wavelets with -dilations

We study Parseval frame wavelets in with matrix dilations of the form , where A is an arbitrary expanding n×n matrix with integer coefficients, such that |detA|=2. We show that each A-MRA admits either Parseval frame wavelets, or Parseval frame bi-wavelets. The minimal number of generators for a Parseval frame associated with an A-MRA (i.e. 1 or 2) is determined in terms of a scaling function. All Parseval frame (bi)wavelets associated with A-MRA's are described. We then introduce new classes of filter induced wavelets and bi-wavelets. It is proved that these new classes strictly contain the classes of all A-MRA Parseval frame wavelets and bi-wavelets, respectively. Finally, we demonstrate a method of constructing all filter induced Parseval frame (bi)wavelets from generalized low-pass filters.     

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.     

Nonstationary Wavelets on them-Sphere for Scattered Data

We construct classes of nonstationary wavelets generated by what we callspherical basis functions,which comprise a subclass of Schoenberg's positive definite functions on them-sphere. The wavelets are intrinsically defined on them-sphere and are independent of the choice of coordinate system. In addition, they may be orthogonalized easily, if desired. We will discuss decomposition, reconstruction, and localization for these wavelets. In the special case of the 2-sphere, we derive an uncertainty principle that expresses the trade-off between localization and the presence of high harmonics—or high frequencies—in expansions in spherical harmonics. We discuss the application of this principle to the wavelets that we construct.     

Periodic wavelets on the <Emphasis Type="Italic">p</Emphasis>-adic Vilenkin group

Using the Walsh-Dirichlet type kernel, we construct periodic wavelets on the p-adic Vilenkin group. These wavelets are similar to the trigonometric wavelets which were introduced by C. K. Chui and H. N. Mhaskar [1]. Results on the corresponding fast algorithms for decomposition and reconstruction are also discussed.     

Uncertainty products of local periodic wavelets

This paper is on the angle–frequency localization of periodic scaling functions and wavelets. It is shown that the uncertainty products of uniformly local, uniformly regular and uniformly stable scaling functions and wavelets are uniformly bounded from above by a constant. Results for the construction of such scaling functions and wavelets are also obtained. As an illustration, scaling functions and wavelets associated with a family of generalized periodic splines are studied. This family is generated by periodic weighted convolutions, and it includes the well‐known periodic B‐splines and trigonometric B‐splines. This revised version was published online in June 2006 with corrections to the Cover Date.     

Construction of optimally conditioned cubic spline wavelets on the interval

The paper is concerned with a construction of new spline-wavelet bases on the interval. The resulting bases generate multiresolution analyses on the unit interval with the desired number of vanishing wavelet moments for primal and dual wavelets. Both primal and dual wavelets have compact support. Inner wavelets are translated and dilated versions of well-known wavelets designed by Cohen, Daubechies, and Feauveau. Our objective is to construct interval spline-wavelet bases with the condition number which is close to the condition number of the spline wavelet bases on the real line, especially in the case of the cubic spline wavelets. We show that the constructed set of functions is indeed a Riesz basis for the space L ² ([0, 1]) and for the Sobolev space H ^s ([0, 1]) for a certain range of s. Then we adapt the primal bases to the homogeneous Dirichlet boundary conditions of the first order and the dual bases to the complementary boundary conditions. Quantitative properties of the constructed bases are presented. Finally, we compare the efficiency of an adaptive wavelet scheme for several spline-wavelet bases and we show a superiority of our construction. Numerical examples are presented for the one-dimensional and two-dimensional Poisson equations where the solution has steep gradients.     

Wavelets for multichannel signals

In this paper, we introduce and investigate multichannel wavelets, which are wavelets for vector fields, based on the concept of full rank subdivision operators. We prove that, like in the scalar and multiwavelet case, the existence of a scaling function with orthogonal integer translates guarantees the existence of a wavelet function, also with orthonormal integer translates. In this context, however, scaling functions as well as wavelets turn out to be matrix-valued functions.     

A New Method of Constructions of Non-Tensor Product Wavelets

In this paper, we give a new method of constructions of non-tensor product wavelets. We start from the one-dimensional scaling functions to directly construct the two-dimensional non-tensor product wavelets. The wavelets constructed by us possess very simple, explicit representations and high regularity, and various symmetry (i.e., axial symmetry, central symmetry, and cyclic symmetry). Using this method, we construct various non-tensor product wavelets and show that there exists a sequence of non-tensor product wavelets with high regularity which tends to the tensor product Shannon wavelet in the L ²-norm.     

On some class of nonseparable continuous wavelet transforms

It is well-known that only a single condition (called the admissibility condition) is sufficient for L ₂-convergence of multiple continuous wavelet transforms (MCWT). However known results suggest that to guarantee the pointwise convergence of MCWT for L _p -functions wavelets should vanish quite rapidly at infinity. In this article, we consider the class of nonseparable multiple wavelets with square-symmetric Fourier transforms. For these wavelets ψ we prove that the admissibility condition and the condition ψ∈L ₁(R ⁿ ) are sufficient for the convergence of corresponding MCWT almost everywhere.     

Wavelet bases of Hermite cubic splines on the interval

In this paper a pair of wavelets are constructed on the basis of Hermite cubic splines. These wavelets are in C¹ and supported on [−1,1]. Moreover, one wavelet is symmetric, and the other is antisymmetric. These spline wavelets are then adapted to the interval [0,1]. The construction of boundary wavelets is remarkably simple. Furthermore, global stability of the wavelet basis is established. The wavelet basis is used to solve the Sturm–Liouville equation with the Dirichlet boundary condition. Numerical examples are provided. The computational results demonstrate the advantage of the wavelet basis. Dedicated to Dr. Charles A. Micchelli on the occasion of his 60th birthday Mathematics subject classifications (2000) 42C40, 41A15, 65L60. Research was supported in part by NSERC Canada under Grants # OGP 121336.     

Numerical solution of Fredholm integral equations of first kind by two-dimensional trigonometric wavelets in holder space <Emphasis Type="Italic">C</Emphasis><Superscript>α</Superscript>([<Emphasis Type="Italic">a,b</Emphasis>])

In this article, we employ trigonometric wavelet bases to numerical solution of Fredholm integral equations of first kind in Holder space. Employment of Galerkin method for trigonometric wavelets in Fredholm integral equations of first kind has resulted in occurrence of two-dimensional trigonometric wavelets. Here, we present the convergence of two-dimensional trigonometric wavelets in numerical solution in Holder space C ^α([a, b]).     

Littlewood-Paley spline wavelets: a simple and efficient tool for signal and image processing in industrial applications

In this work, we present the Littlewood-Paley Spline Wavelets that are much more effective that the orthonormal discrete wavelets, although it cannot apply the multiresolution scheme. These wavelets carry the advantages of the splines functions what allows their implementation through simple numeric algorithms. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Compactly supported box-spline wavelets

A general procedure for constructing multivariate non-tensor-product wavelets that generate an orthogonal decomposition ofL ²(R)^s,s s≥1, is described and applied to yield explicit formulas for compactly supported spline-wavelets based on the multiresolution analysis ofL ²(R)^s 1≤s≤3, generated by any box spline whose direction set constitutes a unimodular matrix. In particular, when univariate cardinal B-splines are considered, the minimally supported cardinal spline-wavelets of Chui and Wang are recovered. A refined computational scheme for the orthogonalization of spaces with compactly supported wavelets is given. A recursive approximation scheme for “truncated” decomposition sequences is developed and a sharp error bound is included. A condition on the symmetry or anti-symmetry of the wavelets is applied to yield symmetric box-spline wavelets. Partially supported by ARO Grant DAAL 03-90-G-0091 Partially supported by NSF Grant DMS 89-0-01345 Partially supported by NATO Grant CRG 900158.     

Numerical stability of biorthogonal wavelet transforms

For orthogonal wavelets, the discrete wavelet and wave packet transforms and their inverses are orthogonal operators with perfect numerical stability. For biorthogonal wavelets, numerical instabilities can occur. We derive bounds for the 2-norm and average 2-norm of these transforms, including efficient numerical estimates if the numberL of decomposition levels is small, as well as growth estimates forL . These estimates allow easy determination of numerical stability directly from the wavelet coefficients. Examples show that many biorthogonal wavelets are in fact numerically well behaved.     

Tight frames and geometric properties of wavelet sets

A construction for providing single dyadic orthonormal wavelets in Euclidean space ℝ^d is given. It is called the general neighborhood mapping construction. The fact that one wavelet is sufficient to generate an orthonormal basis for L²(ℝ^d) is the critical issue. The validity of the construction is proved, and the construction is implemented computationally to provide a host of examples illustrating various geometrical properties of such wavelets in the spectral domain. Because of the inherent complexity of these single orthonormal wavelets, the method is applied to the construction of single dyadic tight frame wavelets, and these tight frame wavelets can be surprisingly simple in nature. The structure of the spectral domains of the wavelets arising from the general neighborhood mapping construction raises a basic geometrical question. There is also the question of whether or not the general neighborhood mapping construction gives rise to all single dyadic orthonormal wavelets. Results are proved giving partial answers to both of these questions. Dedicated to Charles A. Micchelli for his 60th birthday Mathematics subject classification (2000) 42C40. John J. Benedetto: Both authors gratefully acknowledge support from ONR Grant N000140210398. The first named author also gratefully acknowledges support from NSF DMS Grant 0139759.     

Numerical solution of nth‐order integro‐differential equations using trigonometric wavelets

The main aim of this paper is to apply the trigonometric wavelets for the solution of the Fredholm integro‐differential equations of nth‐order. The operational matrices of derivative for trigonometric scaling functions and wavelets are presented and are utilized to reduce the solution of the Fredholm integro‐differential equations to the solution of algebraic equations. Furthermore, we get an estimation of error bound for this method. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

Riesz Bases and Multiresolution Analyses

Recently we found a family of nearly orthonormal affine Riesz bases of compact support and arbitrary degrees of smoothness, obtained by perturbing the one-dimensional Haar mother wavelet using B-splines. The mother wavelets thus obtained are symmetric and given in closed form, features which are generally lacking in the orthogonal case. We also showed that for an important subfamily the wavelet coefficients can be calculated in O(n) steps, just as for orthogonal wavelets. It was conjectured by Aldroubi, and independently by the author, that these bases cannot be obtained by a multiresolution analysis. Here we prove this conjecture. The work is divided into four sections. The first section is introductory. The main feature of the second is simple necessary and sufficient conditions for an affine Riesz basis to be generated by a multiresolution analysis, valid for a large class of mother wavelets. In the third section we apply the results of the second section to several examples. In the last section we show that our bases cannot be obtained by a multiresolution analysis.