An Equivalence Relation between Multiresolution Analysis ofL(R)

This paper is concerned with the development of an equivalence relation between two multiresolution analysis ofL²(R). The relation called unitary equivalence is created by the action of a unitary operator in such a way that the multiresolution structure and the decomposition and reconstruction algorithms remain invariant. A characterization in terms of the scaling functions of the multiresolution analysis is given. Distinct equivalence classes of multiresolution analysis are derived. Finally, we prove that B-splines give rise to nonequivalent examples.     

Generalized low pass filters and MRA frame wavelets

A tight frame wavelet ψ is an L ²(ℝ) function such that {ψ jk(x)} = {2^j/2 ψ(2 ^j x −k), j, k ∈ ℤ},is a tight frame for L ² (ℝ).We introduce a class of “generalized low pass filters” that allows us to define (and construct) the subclass of MRA tight frame wavelets. This leads us to an associated class of “generalized scaling functions” that are not necessarily obtained from a multiresolution analysis. We study several properties of these classes of “generalized” wavelets, scaling functions and filters (such as their multipliers and their connectivity). We also compare our approach with those recently obtained by other authors.     

Symmetric orthonormal scaling functions and wavelets with dilation factor 4

It is well known that in the univariate case, up to an integer shift and possible sign change, there is no dyadic compactly supported symmetric orthonormal scaling function except for the Haar function. In this paper we are concerned with the construction of symmetric orthonormal scaling functions with dilation factor d=4. Several examples of such orthonormal scaling functions are provided in this paper. In particular, two examples of C ¹ orthonormal scaling functions, which are symmetric about 0 and 1/6, respectively, are presented. We will then discuss how to construct symmetric wavelets from these scaling functions. We explicitly construct the corresponding orthonormal symmetric wavelets for all the examples given in this paper. This revised version was published online in June 2006 with corrections to the Cover Date.     

A COMPACTLY SUPPORTED ORTHONORMAL WAVELET OF NON-TENSOR TYPE

Abstract. In this note we construct an example of compactly supported orthonormal wavelets of non-tensor type from a multiresolutlon of     

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.     

p-Adic Haar Multiresolution Analysis and Pseudo-Differential Operators

The notion of p-adic multiresolution analysis (MRA) is introduced. We discuss a “natural” refinement equation whose solution (a refinable function) is the characteristic function of the unit disc. This equation reflects the fact that the characteristic function of the unit disc is a sum of p characteristic functions of mutually disjoint discs of radius p ⁻¹. This refinement equation generates a MRA. The case p=2 is studied in detail. Our MRA is a 2-adic analog of the real Haar MRA. But in contrast to the real setting, the refinable function generating our Haar MRA is 1-periodic, which never holds for real refinable functions. This fact implies that there exist infinity many different 2-adic orthonormal wavelet bases in ℒ²(ℚ₂) generated by the same Haar MRA. All of these new bases are described. We also constructed infinity many different multidimensional 2-adic Haar orthonormal wavelet bases for ℒ²(ℚ₂ ⁿ ) by means of the tensor product of one-dimensional MRAs. We also study connections between wavelet analysis and spectral analysis of pseudo-differential operators. A criterion for multidimensional p-adic wavelets to be eigenfunctions for a pseudo-differential operator (in the Lizorkin space) is derived. We proved also that these wavelets are eigenfunctions of the Taibleson multidimensional fractional operator. These facts create the necessary prerequisites for intensive using our wavelet bases in applications. Our results related to the pseudo-differential operators develop the investigations started in Albeverio et al. (J. Fourier Anal. Appl. 12(4):393–425, 2006).      

Crystallographic Haar Wavelets

(Γ,a)-crystallographic multiwavelets are a finite set of functions Y = { yⁱ}_i=1^L\Psi= \{ \psi ^{i}\}_{i=1}^{L}, which generate an orthonormal basis, a Riesz basis or a Parseval frame for L ²(ℝ ^d ), under the action of a crystallographic group Γ, and powers of an appropriate expanding affine map a, taking the place of the translations and dilations in classical wavelets respectively. Associated crystallographic multiresolution analysis of multiplicity n ((Γ,a)-MRA) are defined in a natural way. A complete characterization of scaling function vectors which generates Haar type (Γ,a)-MRA’s in terms of (Γ,a)-multireptiles is given. Examples of (Γ,a)-MRA crystallographic wavelets of Haar type in dimension 2 and 3 are provided.     

Using the refinement equation for the construction of pre-wavelets III: Elliptic splines

The purpose of this paper is to provide multiresolution analysis, stationary subdivision and pre-wavelet decomposition onL ²(R ^d ) based on a general class of functions which includes polyharmonic B-splines.The work of this author has been partially supported by a DARPA grant.The work of this author has been partially supported by Fondo Nacional de Ciencia y Technologia under Grant 880/89.     

Compactly Supported Tight Frames Associated with Refinable Functions

It is well known that in applied and computational mathematics, cardinal B-splines play an important role in geometric modeling (in computer-aided geometric design), statistical data representation (or modeling), solution of differential equations (in numerical analysis), and so forth. More recently, in the development of wavelet analysis, cardinal B-splines also serve as a canonical example of scaling functions that generate multiresolution analyses of L²(−∞,∞). However, although cardinal B-splines have compact support, their corresponding orthonormal wavelets (of Battle and Lemarie) have infinite duration. To preserve such properties as self-duality while requiring compact support, the notion of tight frames is probably the only replacement of that of orthonormal wavelets. In this paper, we study compactly supported tight frames Ψ={ψ¹,…,ψ^N} for L²(−∞,∞) that correspond to some refinable functions with compact support, give a precise existence criterion of Ψ in terms of an inequality condition on the Laurent polynomial symbols of the refinable functions, show that this condition is not always satisfied (implying the nonexistence of tight frames via the matrix extension approach), and give a constructive proof that when Ψ does exist, two functions with compact support are sufficient to constitute Ψ, while three guarantee symmetry/anti-symmetry, when the given refinable function is symmetric.     

Generalized Multiresolution Analyses with Given Multiplicity Functions

Generalized multiresolution analyses are increasing sequences of subspaces of a Hilbert space ℋ that fail to be multiresolution analyses in the sense of wavelet theory because the core subspace does not have an orthonormal basis generated by a fixed scaling function. Previous authors have studied a multiplicity function m which, loosely speaking, measures the failure of the GMRA to be an MRA. When the Hilbert space ℋ is L ²(ℝ ⁿ ), the possible multiplicity functions have been characterized by Baggett and Merrill. Here we start with a function m satisfying a consistency condition, which is known to be necessary, and build a GMRA in an abstract Hilbert space with multiplicity function m.     

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.     

The Heat Equation for the Generalized Hermite and the Generalized Landau Operators

We give a formula for the one-parameter strongly continuous semigroups ${e^{-tL^{\lambda}}}We give a formula for the one-parameter strongly continuous semigroups e^-tLl{e^{-tL^{\lambda}}} and e^{-t [(A)\tilde]}{e^{-t \tilde{A}}}, t > 0 generated by the generalized Hermite operator L^l, l ? R\{0}{L^{\lambda}, \lambda \in {\bf R}\backslash \{0\}} respectively by the generalized Landau operator ?. These formula are derived by means of pseudo-differential operators of the Weyl type, i.e. Weyl transforms, Fourier-Wigner transforms and Wigner transforms of some orthonormal basis for L ²(R ²ⁿ ) which consist of the eigenfunctions of the generalized Hermite operator and of the generalized Landau operator. Applications to an L ² estimate for the solutions of initial value problems for the heat equations governed by L ^λ respectively ?, in terms of L ^p norm, 1 ≤ p ≤ ∞ of the initial data are given.     

Quaternion-valued admissible wavelets and orthogonal decomposition of L 2(IG(2), ℍ)

A series of admissible wavelets is fixed, which forms an orthonormal basis for the Hilbert space of all the quaternion-valued admissible wavelets. It turns out that their corresponding admissible wavelet transforms give an orthogonal decomposition of L ²(IG(2), ℍ).      

A general construction of nonseparable multivariate orthonormal wavelet bases

The construction of nonseparable and compactly supported orthonormal wavelet bases of L ²(R ⁿ ); n ≥ 2, is still a challenging and an open research problem. In this paper, we provide a special method for the construction of such wavelet bases. The wavelets constructed by this method are dyadic wavelets. Also, we show that our proposed method can be adapted for an eventual construction of multidimensional orthogonal multiwavelet matrix masks, candidates for generating multidimensional multiwavelet bases.      

Approximation by p -Faber polynomials in the weighted Smirnov class Ep( G,ω ) and the Bieberbach polynomials

Let G\subset C be a finite domain with a regular Jordan boundary L . In this work, the approximation properties of a p -Faber polynomial series of functions in the weighted Smirnov class E ^p (G,ω) are studied and the rate of polynomial approximation, for f∈ E ^p ( G,ω) by the weighted integral modulus of continuity, is estimated. Some application of this result to the uniform convergence of the Bieberbach polynomials π _n in a closed domain \overline G with a smooth boundary L is given. February 25, 1999. Date revised: October 20, 1999. Date accepted: May 26, 2000.     

A Characterization for Windowed Fourier Orthonormal Basis with Compact Support

Let g(x) ∈L ²(R) and ğ(ω) be the Fourier transform of g(x). Define g _mn (x) = e ^imx g(x−2πn). In this paper we shall give a sufficient and necessary condition under which {g _mn (x)} constitutes an orthonormal basis of L ²(R) for compactly supported g(ω) or ˘(ω). Received March 25, 1999, Revised November 5, 1999, Accepted September 6, 2000     

Radial Multiresolution, Cuntz Algebras Representations and an Application to Fractals

We study Bernoulli type convolution measures on attractor sets arising from iterated function systems on R. In particular we examine orthogonality for Hankel frequencies in the Hilbert space of square integrable functions on the attractor coming from a radial multiresolution analysis on R³. A class of fractals emerges from a finite system of contractive affine mappings on the zeros of Bessel functions. We have then fractal measures on one hand and the geometry of radial wavelets on the other hand. More generally, multiresolutions serve as an operator theoretic framework for the study of such selfsimilar structures as wavelets, fractals, and recursive basis algorithms. The purpose of the present paper is to show that this can be done for a certain Bessel–Hankel transform. Submitted: February 20, 2008., Accepted: March 6, 2008.     

Integration and approximation in arbitrary dimensions

We study multivariate integration and approximation for various classes of functions of d variables with arbitrary d. We consider algorithms that use function evaluations as the information about the function. We are mainly interested in verifying when integration and approximation are tractable and strongly tractable. Tractability means that the minimal number of function evaluations needed to reduce the initial error by a factor of ɛ is bounded by C(d)ɛ^−p for some exponent p independent of d and some function C(d). Strong tractability means that C(d) can be made independent of d. The ‐exponents of tractability and strong tractability are defined as the smallest powers of ɛ{^-1} in these bounds. We prove that integration is strongly tractable for some weighted Korobov and Sobolev spaces as well as for the Hilbert space whose reproducing kernel corresponds to the covariance function of the isotropic Wiener measure. We obtain bounds on the ‐exponents, and for some cases we find their exact values. For some weighted Korobov and Sobolev spaces, the strong ‐exponent is the same as the ‐exponent for d=1, whereas for the third space it is 2. For approximation we also consider algorithms that use general evaluations given by arbitrary continuous linear functionals as the information about the function. Our main result is that the ‐exponents are the same for general and function evaluations. This holds under the assumption that the orthonormal eigenfunctions of the covariance operator have uniformly bounded L∞ norms. This assumption holds for spaces with shift-invariant kernels. Examples of such spaces include weighted Korobov spaces. For a space with non‐shift‐invariant kernel, we construct the corresponding space with shift-invariant kernel and show that integration and approximation for the non-shift-invariant kernel are no harder than the corresponding problems with the shift-invariant kernel. If we apply this construction to a weighted Sobolev space, whose kernel is non-shift-invariant, then we obtain the corresponding Korobov space. This enables us to derive the results for weighted Sobolev spaces. This revised version was published online in June 2006 with corrections to the Cover Date.