Homogeneous Approximation Property for Multivariate Continuous Wavelet Transforms

The homogeneous approximation property (HAP) for the continuous wavelet transform is useful in practice because it means that the measure of the building area involved in a reconstruction of a function up to some error is essentially invariant under timescale shifts. For the univariate case, it was shown that the pointwise HAP holds if and only if the Fourier transforms of both wavelets and the function to be reconstructed are compactly supported on ??{0}. In this paper, we study the HAP for multivariate wavelet transforms. We show that similar results hold for this case. However, the above condition is only sufficient but not necessary if the dimension of the variable is greater than 1, which is different from the univariate case. We also get a convergence result on the inverse of wavelet transforms, which improves similar results by Daubechies and Holschneider and Tchamitchain.     

Homogeneous approximation property for wavelet frames with matrix dilations

The homogeneous approximation property (HAP) for wavelet frames was studied recently. The HAP is useful in practice since it means that the number of building blocks involved in a reconstruction of a function up to some error is essentially invariant under time‐scale shifts. In this paper, we prove the HAP for wavelet frames generated by admissible wavelet functions with arbitrary translation parameters and a class of dilation matrices. Moreover, we show that the approximation is uniform to some extent whenever wavelet functions satisfy moderate smooth and decaying conditions (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homogeneous Approximation Property for Wavelet Frames with Matrix Dilations,Ⅱ

The homogeneous approximation property (HAP) states that the number of building blocks involved in a reconstruction of a function up to some error is essentially invariant under time-scale shifts. In this paper, we show that every wavelet frame with nice wavelet function and arbitrary expansive dilation matrix possesses the HAP. Our results improve some known ones.     

Continuous and discrete Bessel wavelet transforms

Using the theory of Hankel convolution, continuous and discrete Bessel wavelet transforms are defined. Certain boundedness results and inversion formula for the continuous Bessel wavelet transform are obtained. Important properties of the discrete Bessel wavelet transform are given.     

Homogeneous approximation property for wavelet frames

The homogeneous approximation property (HAP) of wavelet frames is useful in practice since it means that the number of building blocks involved in a reconstruction of f up to some error is essentially invariant under time-scale shifts. In this paper, we show that every wavelet frame generated with functions satisfying some moderate decay conditions possesses the HAP. Our result improves a recent work of Heil and Kutyniok’s. Moreover, for wavelet frames generated with separable time-scale parameters, i.e., wavelet frames of the form

The strong approximation property and the weak bounded approximation property

We show that the strong approximation property (strong AP) (respectively, strong CAP) and the weak bounded approximation property (respectively, weak BCAP) are equivalent for every Banach space. This gives a negative answer to Oja's conjecture. As a consequence, we show that each of the spaces _c0$c_0$ and _?1${?}_1$ has a subspace which has the AP but fails to have the strong AP.     

A classification of continuous wavelet transforms in dimension three

This paper presents a full catalogue, up to conjugacy and subgroups of finite index, of all matrix groups $H<\mathrm{GL}(3,\mathbb{R})$ that give rise to a continuous wavelet transform with associated irreducible quasi-regular representation. For each group in this class, coorbit theory allows to consistently define spaces of sparse signals, and to construct atomic decompositions converging simultaneously in a whole range of these spaces. As an application of the classification, we investigate the existence of compactly supported admissible vectors and atoms for the groups.     

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.     

On the invariant translation approximation property for discrete groups

Recently J.Roe considered the question of whether for a discrete group the reduced group -algebra is the fixed point algebra of Ad acting on the uniform Roe algebra . is said to have the invariant translation approximation property in this case. We consider a slight generalization of this property which, for exact , is equivalent to the operator space approximation property of . We also give a new characterization of exactness and a short proof of the equivalence of exactness of and exactness of for discrete groups.

     

Three-space problems for the bounded compact approximation property

In this paper, the notion of the bounded compact approximation property (BCAP) of a pair [Banach space and its subspace] is used to prove that if X is a closed subspace of L∞ with the BCAP, then L∞/X has the BCAP. We also show that X* has the λ-BCAP with conjugate operators if and only if the pair (X, Y) has the λ-BCAP for each finite codimensional subspace Y∈X. Let M be a closed subspace of X such that M⊥ is complemented in X*. If X has the (bounded) approximation property of order p, then M has the (bounded) approximation property of order p.     

The high frequency behaviour of continuous wavelet transforms

The high frequency behaviour of continuous wavelet transforms is characterized by the number of vanishing moments of the corresponding basic wavelets. As a consequence we give satisfying answers to the following questions of theoretical as well as practical interest:

What are the differences or similarities between transforms to different wavelets?

Why do wavelet transforms react so sensitively to abrupt signal changes ?     

Numerical stability of biorthogonal wavelet transforms

Biorthogonal wavelets are essential tools for numerous practical applications. It is very important that wavelet transforms work numerically stable in floating point arithmetic. This paper presents new results on the worst-case analysis of roundoff errors occurring in floating point computation of periodic biorthogonal wavelet transforms, i.e. multilevel wavelet decompositions and reconstructions. Both of these wavelet algorithms can be realized by matrix–vector products with sparse structured matrices. It is shown that under certain conditions the wavelet algorithms can be remarkably stable. Numerous tests demonstrate the performance of the results.      

Uncertainty principles for the right-sided multivariate continuous quaternion wavelet transform

ABSTRACT

In this paper, we present some new elements of harmonic analysis related to the right-sided multivariate continuous quaternion wavelet transform. The main objective of this article is to introduce the concept of the right-sided multivariate continuous quaternion wavelet transform and investigate its different properties using the machinery of multivariate quaternion Fourier transform. Last, we have proven a number of uncertainty principles for the right-sided multivariate continuous quaternion wavelet transform.     

Application of wavelet transforms to the solution of boundary value problems for linear parabolic equations

A method based on wavelet transforms is proposed for finding weak solutions to initial-boundary value problems for linear parabolic equations with discontinuous coefficients and inexact data. In the framework of multiresolution analysis, the general scheme for finite-dimensional approximation in the regularization method is combined with the discrepancy principle. An error estimate is obtained for the stable approximate solution obtained by solving a set of linear algebraic equations for the wavelet coefficients of the desired solution.     

Two-dimensional quaternion wavelet transform

In this paper we introduce the continuous quaternion wavelet transform (CQWT). We express the admissibility condition in terms of the (right-sided) quaternion Fourier transform. We show that its fundamental properties, such as inner product, norm relation, and inversion formula, can be established whenever the quaternion wavelets satisfy a particular admissibility condition. We present several examples of the CQWT. As an application we derive a Heisenberg type uncertainty principle for these extended wavelets.     

Ideals of homogeneous polynomials and weakly compact approximation property in Banach spaces

We show that a Banach space E has the weakly compact approximation property if and only if each continuous Banach-valued polynomial on E can be uniformly approximated on compact sets by homogeneous polynomials which are members of the ideal of homogeneous polynomials generated by weakly compact linear operators. An analogous result is established also for the compact approximation property.     

A continuous approximation procedure for determining inventory distribution schemas within supply chains

This paper presents an integrated inventory distribution optimization model that simultaneously incorporates the issues of location, production, inventory, and transportation within a supply chain. The objective is to determine the optimal number and size of shipments under varying but commonly practiced production and shipping scenarios. A continuous approximation procedure is proposed to determine the optimal number and size of shipments. Three production and shipping scenarios are investigated and closed form expressions for the optimal number of shipments for each scenario are obtained. A numerical example is presented to demonstrate the usefulness of the model.     

Application of wavelet bases in linear and nonlinear approximation to functions from Besov spaces

Linear and nonlinear approximations to functions from Besov spaces B _{p, q} ^σ ([0, 1]), σ > 0, 1 ≤ p, q ≤ ∞ in a wavelet basis are considered. It is shown that an optimal linear approximation by a D-dimensional subspace of basis wavelet functions has an error of order D ^{-min(σ, σ + 1/2 ? 1/p)} for all 1 ≤ p ≤ ∞ and σ > max(1/p ? 1/2, 0). An original scheme is proposed for optimal nonlinear approximation. It is shown how a D-dimensional subspace of basis wavelet functions is to be chosen depending on the approximated function so that the error is on the order of D ^?σ for all 1 ≤ p ≤ ∞ and σ > max(1/p ? 1/2, 0). The nonlinear approximation scheme proposed does not require any a priori information on the approximated function.     

On dual and three space problems for the compact approximation property

We introduce the properties W^∗D and BW^∗D for the dual space of a Banach space. And then solve the dual problem for the compact approximation property (CAP): if X^∗ has the CAP and the W^∗D, then X has the CAP. Also, we solve the three space problem for the CAP: for example, if M is a closed subspace of a Banach space such that M^⊥ is complemented in X^∗ and X^∗ has the W^∗D, then X has the CAP whenever X/M has the CAP and M has the bounded CAP. Corresponding problems for the bounded compact approximation property are also addressed.     

A novel approach to solve linear systems arising from BEM using fast wavelet transforms

This paper uses Daubechies orthogonal wavelets to change dense and fully populated matrices of boundary element method (BEM) systems into sparse and semi‐banded matrices. Then a novel algorithm based on hierarchical nature of multiresolution analysis is introduced to solving resultant sparse linear systems. This algorithm decomposes NS‐form of transformed parent matrix into descendant systems with reduced sizes and solves them iteratively using GMRES algorithm. Both parts, changing dense matrices to sparse systems and the novel solver, can be added as a black box to the existing BEM codes. Transforming matrices into wavelet space needs less time than saved by solving sparse large systems. Numerical results with a precise study on sensitivity of solution for physical variables to the thresholding parameter, and savings in computer time and memory are presented. Also, the suitable value for thresholding parameter is recommended for elasticity problems. The results indicate that the proposed method is efficient for large problems. Copyright © 2009 John Wiley & Sons, Ltd.