Hyperbolic Wavelet Approximation

We study the multivariate approximation by certain partial sums (hyperbolic wavelet sums) of wavelet bases formed by tensor products of univariate wavelets. We characterize spaces of functions which have a prescribed approximation error by hyperbolic wavelet sums in terms of a K -functional and interpolation spaces. The results parallel those for hyperbolic trigonometric cross approximation of periodic functions [DPT]. October 16, 1995. Date revised: August 28, 1996.     

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)     

Tree wavelet approximations with applications

We construct a tree wavelet approximation by using a constructive greedy scheme (CGS). We define a function class which contains the functions whose piecewise polynomial approximations generated by the CGS have a prescribed global convergence rate and establish embedding properties of this class. We provide sufficient conditions on a tree index set and on bi-orthogonal wavelet bases which ensure optimal order of convergence for the wavelet approximations encoded on the tree index set using the bi-orthogonal wavelet bases. We then show that if we use the tree index set associated with the partition generated by the CGS to encode a wavelet approximation, it gives optimal order of convergence.     

Restricted Nonlinear Approximation

We introduce a new form of nonlinear approximation called restricted approximation . It is a generalization of n -term wavelet approximation in which a weight function is used to control the terms in the wavelet expansion of the approximant. This form of approximation occurs in statistical estimation and in the characterization of interpolation spaces for certain pairs of L _p and Besov spaces. We characterize, both in terms of their wavelet coefficients and also in terms of their smoothness, the functions which are approximated with a specified rate by restricted approximation. We also show the relation of this form of approximation with certain types of thresholding of wavelet coefficients. March 31, 1998. Date accepted: January 28, 1999.     

On the Orthogonality of Frames and the Density and Connectivity of Wavelet Frames

We examine some recent results of Bownik on density and connectivity of the wavelet frames. We use orthogonality (strong disjointness) properties of frame and Bessel sequences, and also properties of Bessel multipliers (operators that map wavelet Bessel functions to wavelet Bessel functions). In addition we obtain an asymptotically tight approximation result for wavelet frames.     

Adaptive wavelet methods for the stochastic Poisson equation

We study the Besov regularity as well as linear and nonlinear approximation of random functions on bounded Lipschitz domains in ? ^d . The random functions are given either (i) explicitly in terms of a wavelet expansion or (ii) as the solution of a Poisson equation with a right-hand side in terms of a wavelet expansion. In the case (ii) we derive an adaptive wavelet algorithm that achieves the nonlinear approximation rate at a computational cost that is proportional to the degrees of freedom. These results are matched by computational experiments.     

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     

Anisotropic Wavelet Bases and Restricted Nonlinear Approximation

Restricted nonlinear approximation is a generalization of n‐term approximation in which a weight function is used to control the terms of the approximant. Here, restricted nonlinear approximation is considered with respect to anisotropic wavelet bases. In particular, characterizations for those functions, which provide a specific convergence rate by restricted nonlinear approximation, are presented.     

Convex and coconvex-probabilistic wavelet approximation

Continuous functions are approximated by wavelet operators. These preserve convexity and r-convexity and transform continuous probability distribution functions into probability distribution functions at the same time preserving certain convexity conditions. The degree of this approximation is estimated by establishing some Jackson type inequalities     

基于小波神经网络的NLAR(p)过程的逼近

小波神经网络是近年来发展起来的一种逼近非线性函数的新型人工神经网络。特别是，正交尺度函数为某函数的小波神经网络更适合于函数逼近。本文在此基础上讨论了小波神经网络对非线性AR(p)过程的逼近。     

Near Best Tree Approximation

Tree approximation is a form of nonlinear wavelet approximation that appears naturally in applications such as image compression and entropy encoding. The distinction between tree approximation and the more familiar n-term wavelet approximation is that the wavelets appearing in the approximant are required to align themselves in a certain connected tree structure. This makes their positions easy to encode. Previous work [4,6] has established upper bounds for the error of tree approximation for certain (Besov) classes of functions. This paper, in contrast, studies tree approximation of individual functions with the aim of characterizing those functions with a prescribed approximation error. We accomplish this in the case that the approximation error is measured in L ₂, or in the case p2, in the Besov spaces B _p ⁰(L _p ), which are close to (but not the same as) L _p . Our characterization of functions with a prescribed approximation order in these cases is given in terms of a certain maximal function applied to the wavelet coefficients.     

基于小波分析的空间机械臂运动规划的最优控制

讨论了空间机械臂系统非完整运动规划的最优控制问题.利用小波分析方法,将离散正交小波函数引入最优控制,由小波级数展开式逼近替代传统的Fourier基函数,提出基于小波分析的最优控制数值算法.仿真结果表明,该方法对求解空间机械臂非完整运动规划问题是有效的.     

Wavelet Approximation of Periodic Functions

We investigate expansions of periodic functions with respect to wavelet bases. Direct and inverse theorems for wavelet approximation in C and L_p norms are proved. For the functions possessing local regularity we study the rate of pointwise convergence of wavelet Fourier series. We also define and investigate the “discreet wavelet Fourier transform” (DWFT) for periodic wavelets generated by a compactly supported scaling function. The DWFT has one important advantage for numerical problems compared with the corresponding wavelet Fourier coefficients: while fast computational algorithms for wavelet Fourier coefficients are recursive, DWFTs can be computed by explicit formulas without any recursion and the computation is fast enough.     

Interpolating Cubic Spline Wavelet Packet on Arbitrary Partitions

In many applications, the splines on an arbitrary partition are very useful. In this paper, a spline wavelet structure is created in the way that it provides a multiresolution approximation of the spline subspaces with arbitrary partition in the space of continuous functions on a finite interval. Based on the wavelet basis and the wavelet packet in this structure, a multi-level interpolation method is developed for decomposing a function into wavelet series and reconstructing it from its wavelet representation.     

Nonlinear approximation using Gaussian kernels

It is well known that nonlinear approximation has an advantage over linear schemes in the sense that it provides comparable approximation rates to those of the linear schemes, but to a larger class of approximands. This was established for spline approximations and for wavelet approximations, and more recently by DeVore and Ron (in press) [2] for homogeneous radial basis function (surface spline) approximations. However, no such results are known for the Gaussian function, the preferred kernel in machine learning and several engineering problems. We introduce and analyze in this paper a new algorithm for approximating functions using translates of Gaussian functions with varying tension parameters. At heart it employs the strategy for nonlinear approximation of DeVore-Ron, but it selects kernels by a method that is not straightforward. The crux of the difficulty lies in the necessity to vary the tension parameter in the Gaussian function spatially according to local information about the approximand: error analysis of Gaussian approximation schemes with varying tension are, by and large, an elusive target for approximators. We show that our algorithm is suitably optimal in the sense that it provides approximation rates similar to other established nonlinear methodologies like spline and wavelet approximations. As expected and desired, the approximation rates can be as high as needed and are essentially saturated only by the smoothness of the approximand.     

Convergence of Riemannian sums of inverse wavelet transforms

We study the approximation of the inverse wavelet transform using Riemannian sums.We show that when the Fourier transforms of wavelet functions satisfy some moderate decay condition,the Riemannian sums converge to the function to be reconstructed as the sampling density tends to infinity.We also study the convergence of the operators introduced by the Riemannian sums.Our result improves some known ones.     

Coiflet interpolation and approximate solutions of elliptic partial differential equations

In this article, we prove a higher order interpolation result for square–integrable functions by using generalized coiflets. Convergence of approximation by using generalized coiflets is shown. Applications to wavelet–Galerkin approximation of elliptic partial differential equations and some numerical examples are also given. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13:303–320, 1997.     

Inversion of the wavelet transform using Riemannian sums

We study the approximation of the inverse wavelet transform using Riemannian sums. For a large class of wavelet functions, we show that the Riemannian sums converge to the original function as the sampling density tends to infinity. When the analysis and synthesis wavelets are the same, we also give some necessary conditions for the Riemannian sums to be convergent.     

On the fast approximation of some nonlinear operators in nonregular wavelet spaces

We focus our attention on the approximation of some nonlinear operators in adapted wavelet spaces. We show the interest of the construction of scaling functions with a large number of zero moments. We present the convergence estimate of an algorithm based on paraproducts for the approximation of nonlinear operators using wavelets connected to scaling functions with zero moments. Numerical tests are performed on univariate examples. This revised version was published online in June 2006 with corrections to the Cover Date.     

Wavelets and framelets from dual pseudo splines

Dual pseudo splines constitute a new class of refinable functions with B-splines as special examples.In this paper,we shall construct Riesz wavelet associated with dual pseudo splines.Furthermore,we use dual pseudo splines to construct tight frame systems with desired approximation order by applying the unitary extension principle.