Construction of Periodic Prolate Spheroidal Wavelets Using Interpolation

Periodic prolate spheroidal wavelets (periodic PS wavelets), based on the periodizaton of the first prolate spheroidal wave function (PSWF), were recently introduced by the authors. Because of localization and other properties, these periodic PS wavelets could serve as an alternative to Fourier series for applications in modeling periodic signals. In this paper, we continue our work with periodic PS wavelets and direct our attention to their construction via interpolation. We show that they have a representation in terms of interpolation with the modified Dirichlet kernel. We then derive a group of formulas of interpolation type based on this representation. These formulas enable one to obtain a simple procedure for the calculation of the periodic PS wavelets and finding expansion coefficients. In particular, they are used to compute filter coefficients for the periodic PS wavelets. This is done for a number of concrete cases.     

Periodic Prolate Spheroidal Wavelets

Prolate spheroidal wavelets (PS wavelets) were recently introduced by the authors. They were based on the first prolate spheroidal wave function (PSWF) and had many desirable properties lacking in other wavelets. In particular, the subspaces belonging to the associated multiresolution analysis (MRA) were shown to be closed under differentiation and translation. In this paper, we introduce periodic prolate spheroidal wavelets. These periodic wavelets are shown to possess properties inherited from PS wavelets such as differentiation and translation. They have the potential for applications in modeling periodic phenomena as an alternative to the usual periodic wavelets as well as the Fourier basis.     

一类周期小波的局部性质

在文献[1]中,陈翰麟等构造了一类具有很好性质的周期小波．我们在这篇论文中进一步研究了该类周期小波,证明了它们在一个周期内具有局部性质．     

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.     

BIVARIATE REAL-VALUED ORTHOGONAL PERIODIC WAVELETS

In this paper, we construct a kind of bivariate real-valued orthogonal periodic wavelets. The corresponding decomposition and reconstruction algorithms involve only 8 terms respectively which are very simple in practical computation. Moreover, the relation between periodic wavelets and Fourier series is also discussed.     

二元实值正交周期Box样条小波

In this paper, we construct a kind of bivariate real-valued orthogonal periodic box-spline wavelets. There are only 4 terms in the two-scale dilation equations. This implies that the corresponding decomposition and reconstruction algorithms involve only 4 terms respectively which are simple in practical computation. The relation between the periodic wavelets and Fourier series is also discussed.     

紧支撑样条小波插值及其应用

基于紧支撑样条小波函数插值与定积分的思想,给出了由紧支撑样条小波插值函数构造数值积分公式的方法．并将该方法应用于二次、三次、四次和五次紧支撑样条小波函数,得到了相应的数值积分公式．最后,通过数值例子验证,发现该方法得到的数值积分公式是准确的,且具有较高精度．     

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.     

A Family of Approximation Formulas and some Applications

The general scheme, suggested in [1] using a basis of an infinite-dimensional space and allowing to construct finite-dimensional orthogonal systems and interpolation formulas, is improved in the paper. This results particularly in a generalization of the well-known scheme by which periodic interpolatory wavelets are constructed. A number of systems which do not satisfy all the conditions for multiresolution analysis but have some useful properties are introduced and investigated.

Starting with general constructions in Hilbert spaces, we give a more careful consideration to the case connected with the classic Fourier basis.

Convergence of expansions which are similar to partial sums of the summation method of Fourier series, as well as convergence of interpolation formulas are considered.

Some applications to fast calculation of Fourier coefficients and to solution of integrodifferential equations are given. The corresponding numerical results have been obtained by means of MATHEMATICA 3.0 system.     

一类向量值正交小波的设计与性质

引入整数因子伸缩的向量值正交小波与向量值小波包的概念.运用仿酉向量滤波器理论和矩阵理论,给出具有整数因子伸缩的向量值正交小波存在的充要条件.提供了紧支撑向量值正交的构建算法,给出了相应的构建算例.利用时频分析方法与算子理论,刻画了一类向量值正交小波包的性质,得到了整数伸缩的向量值小波包的正交公式.     

Interpolating-orthogonal wavelet systems

Based upon Meyer wavelets, new systems of periodic wavelets and wavelets on the whole axis are constructed; these systems are orthogonal and interpolating simultaneously. Estimates of the errors of approximation of different classes of smooth functions by these wavelets are obtained.     

Periodic Wavelets from Scratch

Periodic wavelets can be constructed from most standard wavelets by periodization. In this work we first derive some of their properties and then construct the periodic wavelets directly from their Fourier series without reference to standard wavelets. Several examples are given some of which are not constructable from the usual wavelets on the real line.     

PERIODIC CARDINAL INTERPOLATORY WAVELETS

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Periodic dyadic wavelets and coding of fractal functions

Recently, using the Walsh-Dirichlet type kernels, the first author has defined periodic dyadic wavelets on the positive semiaxis which are similar to the Chui-Mhaskar trigonometric wavelets. In this paper we generalize this construction and give examples of applications of periodic dyadic wavelets for coding the Riemann, Weierstrass, Schwarz, van der Waerden, Hankel, and Takagi fractal functions.     

Poisson Wavelets on the Sphere

In this article we summarize the basic formulas of wavelet analysis with the help of Poisson wavelets on the sphere. These wavelets have the nice property that all basic formulas of wavelet analysis as reproducing kernels, etc. may be expressed simply with the help of higher degree Poisson wavelets. This makes them numerically attractive for applications in geophysical modeling.     

A General Framework of Multivariate Wavelets with Duals

A comprehensive development of multivariate wavelets along with their duals is presented in this paper. The basic ingredients, such as duality relations, reconstruction and decomposition formulas, and the notion of infinite direct sums, are formulated and established in the general nonorthogonal and multivariate setting. Special emphases include duality criteria and stability conditions. As an application, new results are contributed, particularly in dual wavelets, to the existing literature on low-dimensional wavelets. In addition, the special case when the dilation matrix has determinant 2 is studied in some detail.     

Construction of Biorthogonal Discrete Wavelet Transforms Using Interpolatory Splines

We present a new family of biorthogonal wavelet and wavelet packet transforms for discrete periodic signals and a related library of biorthogonal periodic symmetric waveforms. The construction is based on the superconvergence property of the interpolatory polynomial splines of even degrees. The construction of the transforms is performed in a “lifting” manner that allows more efficient implementation and provides tools for custom design of the filters and wavelets. As is common in lifting schemes, the computations can be carried out “in place” and the inverse transform is performed in a reverse order. The difference with the conventional lifting scheme is that all the transforms are implemented in the frequency domain with the use of the fast Fourier transform. Our algorithm allows a stable construction of filters with many vanishing moments. The computational complexity of the algorithm is comparable with the complexity of the standard wavelet transform. Our scheme is based on interpolation and, as such, it involves only samples of signals and it does not require any use of quadrature formulas. In addition, these filters yield perfect frequency resolution.     

Construction and decomposition of biorthogonal vector-valued wavelets with compact support

In this article, we introduce vector-valued multiresolution analysis and the biorthogonal vector-valued wavelets with four-scale. The existence of a class of biorthogonal vector-valued wavelets with compact support associated with a pair of biorthogonal vector-valued scaling functions with compact support is discussed. A method for designing a class of biorthogonal compactly supported vector-valued wavelets with four-scale is proposed by virtue of multiresolution analysis and matrix theory. The biorthogonality properties concerning vector-valued wavelet packets are characterized with the aid of time–frequency analysis method and operator theory. Three biorthogonality formulas regarding them are presented.     

Two-dimension periodic cardinal interpolatory wavelets

In this paper, we construct two-dimension periodic interpolatory scaling function and wavelets from a periodic function g(x₁, x₂), whose Fourier coefficients are positive, and obtain some properties of scaling functions and wavelets.     

Two-dimension periodic cardinal interpolatory wavelets

In this paper, we construct two-dimension periodic interpolatory scaling function and wavelets from a periodic function g(x₁, x₂), whose Fourier coefficients are positive, and obtain some properties of scaling functions and wavelets.