Measures, densities and diameters of frequency bands of scaling functions and wavelets

We investigate the measures of frequency bands of scaling functions and wavelets, and solve an open problem. Furthermore, we give conditions for which the measures of frequency bands of scaling functions are less than and the measures of frequency bands of wavelets are less than 4π. We also discuss the densities around the origin and the diameters of frequency bands of scaling functions and wavelets, and show that there are essential differences between the single and multidimensional cases.     

Estimating copula densities through wavelets

Wavelet analysis is used to construct a rank-based estimator of a copula density. The procedure, which can be easily implemented with ready-to-use wavelet packages, is based on an algorithm that handles boundary effects automatically. The resulting estimator provides a non-parametric benchmark for the selection of a parametric copula family. From a theoretical point of view, the estimation procedure is shown to be optimal in the minimax sense on a large functional class of regular copula densities. The approach is illustrated with actuarial and financial data.     

Smoothing minimally supported frequency wavelets: Part II

The main purpose of this paper is to give a procedure to “mollify” the low-pass filters of a large number ofMinimally Supported Frequency (MSF) wavelets so that the smoother functions obtained in this way are also low-pass filters for an MRA. Hence, we are able to approximate (in the L ² -norm) MSF wavelets by wavelets with any desired degree of smoothness on the Fourier transform side. Although the MSF wavelets we consider are bandlimited, this may not be true for their smooth approximations. This phenomena is related to the invariant cycles under the transformation x ↦2x (mod2π). We also give a characterization of all low-pass filters for MSF wavelets. Throughout the paper new and interesting examples of wavelets are described.     

Smooth orthonormal wavelets with exponential decay in both time and frequency domains

This article presents the existence of a non-trivial smooth orthonormal wavelet with exponential decay in both time and frequency domains. Meanwhile, we also show that the Meyer wavelets are not in the Schwartz class.     

Band-Limited Wavelets

Using the harmonic method,we get a class of more general band-limited wavelet,which was got by complicate operator interpolation method before.Our result is alittle better than the result by operator interpolation method.The fast band-limitedwavelet transform shall be given in another paper.     

多频率小波

通过方向多分辨分析把由一个函数生成的多频率小波推广到由有限个函数生成的多频率小波，给出由函数φ1,…，φn，…ψn(2^j1 ^j2-1)的平移生成Vj(1)空间的Riesz基的充分必要条件，同时给出该小波的分解式。     

On divergence-free wavelets

This paper is concerned with the construction of compactly supported divergence-free vector wavelets. Our construction is based on a large class of refinable functions which generate multivariate multiresolution analyses which includes, in particular, the non tensor product case.For this purpose, we develop a certain relationship between partial derivatives of refinable functions and wavelets with modifications of the coefficients in their refinement equation. In addition, we demonstrate that the wavelets we construct form a Riesz-basis for the space of divergence-free vector fields.Work supported by the Deutsche Forschungsgemeinschaft in the Graduiertenkolleg Analyse und Konstruktion in der Mathematik at the RWTH Aachen.     

On trigonometric wavelets

Wavelets in terms of sine and cosine functions are constructed for decomposing 2π-periodic square-integrable functions into different octaves and for yielding local information within each octave. Results on a simple mapping into the approximate sample space, order of approximation of this mapping, and pyramid algorithms for decomposition and reconstruction are also discussed.     

Generalized spline wavelets

A generalized multiresolution of multiplicityr, generated byr linearly independent spline functions with multiple knots, is introduced. With the help of the autocorrelation symbol and the two-scale symbol of the scaling functions, spline wavelets with multiple knots can be completely characterized. New decomposition and reconstruction algorithms, based on the Fourier technique, are presented.     

Optimally matched wavelets

Several applications require to retrieve a certain pattern from a signal where the actual scaling of the pattern is not known before. We consider applications like evaluation of mass spectrograms, detection of component wearout by observing the current of an engine, detection of pollutions of rotor spinning machines, decomposition of (audio) signals into time‐frequency atoms. We try to solve this problem with discrete wavelet transforms where the wavelet function is constructed to match the given pattern. An approach for a biorthogonal basis constructed by the lifting scheme is presented. This method is rather simple and fast and allows the choice of the smoothness at least of the primal wavelet in a very natural way. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Regularity of boundary wavelets

The conventional way of constructing boundary functions for wavelets on a finite interval is by forming linear combinations of boundary-crossing scaling functions. Desirable properties such as regularity (i.e. continuity and approximation order) are easy to derive from corresponding properties of the interior scaling functions. In this article we focus instead on boundary functions defined by recursion relations. We show that the number of boundary functions is uniquely determined, and derive conditions for determining regularity from the recursion coefficients. We show that there are regular boundary functions which are not linear combinations of shifts of the underlying scaling functions.     

A construction of wavelets

In this note, given a multiresolution analysis, we construct a class of four-coefficient scaling filters using only elementary algebraic operations. The method of construction also reveals that the sum conditions ∑a _even=∑a _odd=1 can also be verified without referring to the vanishing moment property.     

Basic properties of wavelets

A wavelet multiplier is a function whose product with the Fourier transform of a wavelet is the Fourier transform of a wavelet. We characterize the wavelet multipliers, as well as the scaling function multipliers and low pass filter multipliers. We then prove that if the set of all wavelet multipliers acts on the set of all MRA wavelets, the orbits are the sets of all MRA wavelets whose Fourier transforms have equal absolute values, and these are also equal to the sets, of all MRA wavelets with the corresponding scaling functions having the same absolute values of their Fourier transforms. As an application of these techniques, we prove that the set of MRA wavelets is arcwise connected in L²(R). Dedicated to Eugene Fabes The Wutam Consortium     

On interpolatory divergence-free wavelets

We construct interpolating divergence-free multiwavelets based on cubic Hermite splines. We give characterizations of the relevant function spaces and indicate their use for analyzing experimental data of incompressible flow fields. We also show that the standard interpolatory wavelets, based on the Deslauriers-Dubuc interpolatory scheme or on interpolatory splines, cannot be used to construct compactly supported divergence-free interpolatory wavelets.

     

Tight frames of multidimensional wavelets

In this paper we deal with multidimensional wavelets arising from a multiresolution analysis with an arbitrary dilation matrix A, namely we have scaling equations $$\varphi ^s (x) = \sum\limits_{k \in \mathbb{Z}^n } {h_k^s \sqrt {|\det A|} \varphi ^1 } (Ax - k) for s = 1, \ldots ,q,$$ where ?¹ is a scaling function for this multiresolution and ?², …, ?^q (q=|det A |) are wavelets. Orthogonality conditions for ?¹, …, ?^q naturally impose constraints on the scaling coefficients $\{ h_k^s \} _{k \in \mathbb{Z}^n }^{s = 1, \ldots ,q} $ , which are then called the wavelet matrix. We show how to reconstruct functions satisfying the scaling equations above and show that ?², …, ?^q always constitute a tight frame with constant 1. Furthermore, we generalize the sufficient and necessary conditions of orthogonality given by Lawton and Cohen to the case of several dimensions and arbitrary dilation matrix A.     

Orthogonal decompositions for wavelets

Decompositions of Hilbert spaces in terms of reducing subspaces for wavelets operators, as well decompositions of these operators themselves, are investigated. In particular, it is shown on which reducing subspaces these operators act as bilateral shifts of multiplicity 1. We also exhibit the unitary transformation that performs the unitary equivalence between restrictions of them to appropriate reducing subspaces.     

On wavelets inL 1

In this paper we give a method to characterize the smoothness of functions inL ₁ by anr-regular multiresolution analysis and its derivatives.This project is supported by Zhejiang Provincial Natural Science Foundation and the Special Program of China.