Monotone and probabilistic wavelet approximation

Continuous functions are approximated by wavelet operators. These preserve mone tonicity and transform continuous probability distribution functions into probability distribution functions. The degree of this approximation is estimated by establishing some Jackson type inequalities     

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

Multivariate probabilistic approximation in wavelet structure

Letotherwiseand F(x,y).be a continuous distribution function on R~2.Then there exist linear wavelet operators L_n(F,x,y)which are also distribution functionand where the defining them mother wavelet is(x,y).These approximate F(x,y)in thesupnorm.The degree of this approximation is estimated by establishing a Jackson typeinequality.Furthermore we give generalizations for the case of a mother wavelet ≠,whichis just any distribution function on R~2,also we extend these results in R~r,r>2.     

Haar wavelet approximation for magnetohydrodynamic flow equations

This study proposes Haar wavelet (HW) approximation method for solving magnetohydrodynamic flow equations in a rectangular duct in presence of transverse external oblique magnetic field. The method is based on approximating the truncated double Haar wavelets series. Numerical solution of velocity and induced magnetic field is obtained for steady-state, fully developed, incompressible flow for a conducting fluid inside the duct. The calculations show that the accuracy of the Haar wavelet solutions is quite good even in the case of a small number of grid points. The HW approximation method may be used in a wide variety of high-order linear partial differential equations. Application of the HW approximation method showed that it is reliable, simple, fast, least computation at costs and flexible.     

Robust designs for Haar wavelet approximation models

In this paper, we discuss the construction of robust designs for heteroscedastic wavelet regression models when the assumed models are possibly contaminated over two different neighbourhoods: G ₁ and G ₂ . Our main findings are: (1) A recursive formula for constructing D‐optimal designs under G ₁ ; (2) Equivalency of Q‐optimal and A‐optimal designs under both G ₁ and G ₂ ; (3) D‐optimal robust designs under G ₂ ; and (4) Analytic forms for A‐ and Q‐optimal robust design densities under G ₂ . Several examples are given for the comparison, and the results demonstrate that our designs are efficient. Copyright © 2010 John Wiley & Sons, Ltd.     

Homogeneous approximation property for continuous wavelet transforms

The homogeneous approximation property (HAP) for frames is useful in practice and has been developed recently. In this paper, we study the HAP for the continuous wavelet transform. We show that every pair of admissible wavelets possesses the HAP in L² sense, while it is not true in general whenever pointwise convergence is considered. We give necessary and sufficient conditions for the pointwise HAP to hold, which depends on both wavelets and functions to be reconstructed.     

Nonlinear wavelet approximation and the space C P α

We obtain a Jackson estimate for nonlinear wavelet approximation in the space C _p ^α and some results about the interpolation for C _p ^α     

Biorthogonal wavelet approximation for the coupling of FEM-BEM

Summary. We apply multiscale methods to the coupling of finite and boundary element methods to solve an exterior Dirichlet boundary value problem for the two dimensional Poisson equation. Adopting biorthogonal wavelet matrix compression to the boundary terms with N degrees of freedom, we show that the resulting compression strategy fits the optimal convergence rate of the coupling Galerkin methods, while the number of nonzero entries in the corresponding stiffness matrices is considerably smaller than . Received December 3, 1999 / Revised version received September 22, 2000 / Published online December 18, 2001     

ON THE DEGREE OF APPROXIMATION BY WAVELET EXPANSIONS

In this paper we estimate the degree of approximation of wavelet expansions. Our result shows that the degree has the exponential decay for function f(x) ∈ L2 continuous in a finite interval (a,b) which is much superior to those of approximation by polynomial operators and by expansions of classical orthogonal series.     

Nonlinear approximation schemes associated with nonseparable wavelet bi-frames

In the present paper, we study nonlinear approximation properties of multivariate wavelet bi-frames. For a certain range of parameters, the approximation classes associated with best N-term approximation are determined to be Besov spaces and thresholding the wavelet bi-frame expansion realizes the approximation rate. Our findings extend results about dyadic wavelets to more general scalings. Finally, we verify that the required linear independence assumption is satisfied for particular families of nondyadic wavelet bi-frames in arbitrary dimensions.     

The wavelet transform on Sobolev spaces and its approximation properties

Summary We extend the continuous wavelet transform to Sobolev spacesH ^s() for arbitrary reals and show that the transformed distribution lies in the fiber spaces . This generalisation of the wavelet transform naturally leads to a unitary operator between these spaces.Further the asymptotic behaviour of the transforms ofL ₂-functions for small scaling parameters is examined. In special cases the wevelet transform converges to a generalized derivative of its argument. We also discuss the consequences for the discrete wavelet transform arising from this property. Numerical examples illustrate the main result.Supported by the Deutsche Forschungsgemeinschaft under grant Lo 310/2-4     

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)     

Bootstrap approximation of wavelet estimates in a semiparametric regression model

The inference for the parameters in a semiparametric regression model is studied by using the wavelet and the bootstrap methods. The bootstrap statistics are constructed by using Efron＇s resampling technique, and the strong uniform convergence of the bootstrap approximation is proved. Our results can be used to construct the large sample confidence intervals for the parameters of interest. A simulation study is conducted to evaluate the finite-sample performance of the bootstrap method and to compare it with the normal approximation-based method.     

Discretization algorithm for fractional order integral by Haar wavelet approximation

A discretization algorithm is proposed by Haar wavelet approximation theory for the fractional order integral. In this paper, the integration time is divided into two parts, one presents the effect of the past sampled data, calculated by the iterative method, and the other presents the effect of the recent sampled data at a fixed time interval, calculated by the Haar wavelet. This method can reduce the amount of the stored data effectively and be applied to the design of discrete-time fractional order PID controllers. Finally, several numerical examples and simulation results are given to illustrate the validity of this discretization algorithm.     

Restricted non-linear approximation in sequence spaces and applications to wavelet bases and interpolation

Restricted non-linear approximation is a type of N-term approximation where a measure ν on the index set (rather than the counting measure) is used to control the number of terms in the approximation. We show that embeddings for restricted non-linear approximation spaces in terms of weighted Lorentz sequence spaces are equivalent to Jackson and Bernstein type inequalities, and also to the upper and lower Temlyakov property. As applications we obtain results for wavelet bases in Triebel–Lizorkin spaces by showing the Temlyakov property in this setting. Moreover, new interpolation results for Triebel–Lizorkin and Besov spaces are obtained.     

Solving linear systems using wavelet compression combined with Kronecker product approximation

Discrete wavelet transform approximation is an established means of approximating dense linear systems arising from discretization of differential and integral equations defined on a one-dimensional domain. For higher dimensional problems, approximation with a sum of Kronecker products has been shown to be effective in reducing storage and computational costs. We have combined these two approaches to enable solution of very large dense linear systems by an iterative technique using a Kronecker product approximation represented in a wavelet basis. Further approximation of the system using only a single Kronecker product provides an effective preconditioner for the system. Here we present our methods and illustrate them with some numerical examples. This technique has the potential for application in a range of areas including computational fluid dynamics, elasticity, lubrication theory and electrostatics. AMS subject classification 65F10, 65T60, 65F30 Judith M. Ford: This author was supported by EPSRC Postdoctoral Research Fellowship ref: GR/R95982/01. Current address: Royal Liverpool Children's NHS Trust, Liverpool, L12 2AP. Eugene E. Tyrtyshnikov: This author was supported by the Russian Fund of Basic Research (grant 02-01-00590) and Science Support Foundation.