Stability and Independence for Multivariate Refinable Distributions

Due to their so-called time-frequency localization properties, wavelets have become a powerful tool in signal analysis and image processing. Typical constructions of wavelets depend on the stability of the shifts of an underlying refinable function. In this paper, we derive necessary and sufficient conditions for the stability of the shifts of certain compactly supported refinable functions. These conditions are in terms of the zeros of the refinement mask. Our results are actually applicable to more general distributions which are not of function type, if we generalize the notion of stability appropriately. We also provide a similar characterization of the (global) linear independence of the shifts. We present several examples illustrating our results, as well as one example in which known results on box splines are derived using the theorems of this paper.     

紧支集分布和可加细向量的整平移的线性相关性

In this paper, the global and local linear independence of any compactly supported distributions by using time domain spaces ,and of refinable vectors by invariant linear spaces are investigated.     

Construction of biorthogonal wavelets from pseudo-splines

Pseudo-splines constitute a new class of refinable functions with B-splines, interpolatory refinable functions and refinable functions with orthonormal shifts as special examples. Pseudo-splines were first introduced by Daubechies, Han, Ron and Shen in [Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14(1) (2003), 1–46] and Selenick in [Smooth wavelet tight frames with zero moments, Appl. Comput. Harmon. Anal. 10(2) (2001) 163–181], and their properties were extensively studied by Dong and Shen in [Pseudo-splines, wavelets and framelets, 2004, preprint]. It was further shown by Dong and Shen in [Linear independence of pseudo-splines, Proc. Amer. Math. Soc., to appear] that the shifts of an arbitrarily given pseudo-spline are linearly independent. This implies the existence of biorthogonal dual refinable functions (of pseudo-splines) with an arbitrarily prescribed regularity. However, except for B-splines, there is no explicit construction of biorthogonal dual refinable functions with any given regularity. This paper focuses on an implementable scheme to derive a dual refinable function with a prescribed regularity. This automatically gives a construction of smooth biorthogonal Riesz wavelets with one of them being a pseudo-spline. As an example, an explicit formula of biorthogonal dual refinable functions of the interpolatory refinable function is given.     

Stability and independence of the shifts of finitely many refinable functions

Typical constructions of wavelets depend on the stability of the shifts of an underlying refinable function. Unfortunately, several desirable properties are not available with compactly supported orthogonal wavelets, e.g., symmetry and piecewise polynomial structure. Presently, multiwavelets seem to offer a satisfactory alternative. The study of multiwavelets involves the consideration of the properties of several (simultaneously) refinable functions. In Section 2 of this article, we characterize stability and linear independence of the shifts of a finite refinable function set in terms of the refinement mask. Several illustrative examples are provided. The characterizations given in Section 2 actually require that the refinable functions be minimal in some sense. This notion of minimality is made clear in Section 3, where we provide sufficient conditions on the mask to ensure minimality. The conditions are shown to be necessary also under further assumptions on the refinement mask. An example is provided illustrating how the software package MAPLE can be used to investigate at least the case of two simultaneously refinable functions.     

A new factorization technique of the matrix mask of univariate refinable functions

Summary. A univariate compactly supported refinable function can always be written as the convolution product , with the B-spline of order k,f a compactly supported distribution, and k the approximation orders provided by the underlying shift-invariant space . Factorizations of univariate refinable vectors were also studied and utilized in the literature. One of the by-products of this article is a rigorous analysis of that factorization notion, including, possibly, the first precise definition of that process. The main goal of this article is the introduction of a special factorization algorithm of refinable vectors that generalizes the scalar case as closely (and unexpectedly) as possible: the original vector is shown to be `almost' in the form , with F still compactly supported and refinable, andk the approximation order of . The algorithm guarantees F to retain the possible favorable properties of , such as the stability of the shifts of and/or the polynomiality of the mask symbol. At the same time, the theory and the algorithm are derived under relatively mild conditions and, in particular, apply to whose shifts are not stable, as well as to refinable vectors which are not compactly supported. The usefulness of this specific factorization for the study of the smoothness of FSI wavelets (known also as `multiwavelets' and `multiple wavelets') is explained. The analysis invokes in an essential way the theory of finitely generated shift-invariant (FSI) spaces, and, in particular, the tool of superfunction theory. Received June 10, 1998 / Revised version received June 14, 1999 / Published online August 2, 2000     

Characterization of compactly supported refinable splines with integer matrix

Let M be an integer matrix with absolute values of all its eigenvalues being greater than 1. We give a characterization of compactly supported M-refinable splines f and the conditions that the shifts of f form a Riesz basis.     

Approximation of periodic functions using mup s (x)

The spaces of linear combinations of the shifts of a compactly supported solution of a functional-differential equation are considered. It is proved that they are asymptotically extremal for approximating, in the norm of L ₂, functions from the classes $ \tilde W_2^r $ .     

Supports of Representations in the Cohen Class

In this paper we consider a version of the uncertainty principle concerning limitations on the supports of time-frequency representations in the Cohen class. In particular we obtain various classes of kernels with the property that the corresponding representations of non trivial signals cannot be compactly supported. As an application of our results we show that a linear partial differential operator applied to the Wigner distribution of a function f≠0 in the Schwartz class cannot produce a compactly supported function.     

Generalized sampling reconstruction from Fourier measurements using compactly supported shearlets

In this paper we study the general reconstruction of a compactly supported function from its Fourier coefficients using compactly supported shearlet systems. We assume that only finitely many Fourier samples of the function are accessible and based on this finite collection of measurements an approximation is sought in a finite dimensional shearlet reconstruction space. We analyze this sampling and reconstruction process by a recently introduced method called generalized sampling. In particular by studying the stable sampling rate of generalized sampling we then show stable recovery of the signal is possible using an almost linear rate. Furthermore, we compare the result to the previously obtained rates for wavelets.     

Piecewise constant positive potentials with practically the same fixed energy phase shifts

It has recently been shown that spherically symmetric potentials of finite range are uniquely determined by the part of their phase shifts at a fixed energy level k² > 0. However, numerical experiments show that two quite different potentials can produce almost identical phase shifts. It has been guessed by physicists that such examples are possible only for “less physical” oscillating and changing sign potentials. In this note it is shown that the above guess is incorrect: we give examples of four positive spherically symmetric compactly supported quite different potentials having practically identical phase shifts. The note also describes a hybrid stochastic deterministic method for global minimization for the construction of these potentials.     

Compactly Supported Frames for Decomposition Spaces

In this article we study a construction of compactly supported frame expansions for decomposition spaces of Triebel-Lizorkin type and for the associated modulation spaces. This is done by showing that finite linear combinations of shifts and dilates of a single function with sufficient decay in both direct and frequency space can constitute a frame for Triebel-Lizorkin type spaces and the associated modulation spaces. First, we extend the machinery of almost diagonal matrices to Triebel-Lizorkin type spaces and the associated modulation spaces. Next, we prove that two function systems which are sufficiently close have an almost diagonal “change of frame coefficient” matrix. Finally, we approximate to an arbitrary degree an already known frame for Triebel-Lizorkin type spaces and the associated modulation spaces with a single function with sufficient decay in both direct and frequency space.     

Compactly supported (bi)orthogonal wavelets generated by interpolatory refinable functions

This paper provides several constructions of compactly supported wavelets generated by interpolatory refinable functions. It was shown in [7] that there is no real compactly supported orthonormal symmetric dyadic refinable function, except the trivial case; and also shown in [10,18] that there is no compactly supported interpolatory orthonormal dyadic refinable function. Hence, for the dyadic dilation case, compactly supported wavelets generated by interpolatory refinable functions have to be biorthogonal wavelets. The key step to construct the biorthogonal wavelets is to construct a compactly supported dual function for a given interpolatory refinable function. We provide two explicit iterative constructions of such dual functions with desired regularity. When the dilation factors are larger than 3, we provide several examples of compactly supported interpolatory orthonormal symmetric refinable functions from a general method. This leads to several examples of orthogonal symmetric (anti‐symmetric) wavelets generated by interpolatory refinable functions. This revised version was published online in June 2006 with corrections to the Cover Date.     

Covariance Tapering for Interpolation of Large Spatial Datasets

Interpolation of a spatially correlated random process is used in many scientific areas. The best unbiased linear predictor, often called a kriging predictor in geostatistical science, requires the solution of a (possibly large) linear system based on the covariance matrix of the observations. In this article, we show that tapering the correct covariance matrix with an appropriate compactly supported positive definite function reduces the computational burden significantly and still leads to an asymptotically optimal mean squared error. The effect of tapering is to create a sparse approximate linear system that can then be solved using sparse matrix algorithms. Monte Carlo simulations support the theoretical results. An application to a large climatological precipitation dataset is presented as a concrete and practical illustration.     

A remark on the finite number of particles effect in Monte Carlo methods for kinetic equations

Monte Carlo methods are the most popular methods for solving problems in kinetic theory [2, 5]. In this short remark we emphasize some of the side effects due to the use of conservative methods over a finite number of statistical samples (particles) in the simulation. The most relevant aspect is that the steady states of the system are compactly supported and thus they cannot be Maxwellian (or any other non compactly supported statistics) unless the number of particles goes to infinity. These aspects are studied numerically with the help of a simple one-dimensional space homogeneous kinetic model. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dualizable Shearlet Frames and Sparse Approximation

Shearlet systems have been introduced as directional representation systems, which provide optimally sparse approximations of a certain model class of functions governed by anisotropic features while allowing faithful numerical realizations by a unified treatment of the continuum and digital realm. They are redundant systems, and their frame properties have been extensively studied. In contrast to certain band-limited shearlets, compactly supported shearlets provide high spatial localization but do not constitute Parseval frames. Thus reconstruction of a signal from shearlet coefficients requires knowledge of a dual frame. However, no closed and easily computable form of any dual frame is known. In this paper, we introduce the class of dualizable shearlet systems, which consist of compactly supported elements and can be proved to form frames for \(L^2({\mathbb {R}}^2)\). For each such dualizable shearlet system, we then provide an explicit construction of an associated dual frame, which can be stated in closed form and is efficiently computed. We also show that dualizable shearlet frames still provide near optimal sparse approximations of anisotropic features.     

Local bases for refinable spaces

We provide a new representation of a refinable shift invariant space with a compactly supported generator, in terms of functions with a special property of homogeneity. In particular, these functions include all the homogeneous polynomials that are reproducible by the generator, which links this representation to the accuracy of the space. We completely characterize the class of homogeneous functions in the space and show that they can reproduce the generator. As a result we conclude that the homogeneous functions can be constructed from the vectors associated to the spectrum of the scale matrix (a finite square matrix with entries from the mask of the generator). Furthermore, we prove that the kernel of the transition operator has the same dimension as the kernel of this finite matrix. This relation provides an easy test for the linear independence of the integer translates of the generator. This could be potentially useful in applications to approximation theory, wavelet theory and sampling.

     

Dependence Relations Among the Shifts of a Multivariate Refinable Distribution

Refinable functions are an intrinsic part of subdivision schemes and wavelet constructions. The relevant properties of such functions must usually be determined from their refinement masks. In this paper, we provide a characterization of linear independence for the shifts of a multivariate refinable vector of distributions in terms of its (finitely supported) refinement mask. March 14, 1998. Dates revised: February 3, 1999 and August 6, 1999. Date accepted: November 16, 1999.     

Smooth Compactly Supported Solutions of Some Underdetermined Elliptic PDE,with Gluing Applications

We give sufficient conditions for some underdetermined elliptic PDE of any order to construct smooth compactly supported solutions. In particular we show that two smooth elements in the kernel of certain underdetermined linear elliptic operators P can be glued in a chosen region in order to obtain a new smooth solution. This new solution is exactly equal to the initial elements outside the gluing region. This result completely contrasts with the usual unique continuation for determined or overdetermined elliptic operators. As a corollary we obtain compactly supported solutions in the kernel of P and also solutions vanishing in a chosen relatively compact open region. We apply the result for natural geometric and physics contexts such as divergence free fields or TT-tensors.     

Biorthogonal bases of compactly supported wavelets

Orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide. We show here that under fairly general conditions, exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual Riesz bases of compactly supported wavelets. We give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions, and we present a sufficient conditions for the decay of their Fourier transforms. We study the regularity of these biorthogonal bases. We provide several families of examples, all symmetric (corresponding to “linear phase” filters). In particular we can construct symmetric biorthogonal wavelet bases with arbitraily high preassigned regularity; we also show how to construct symmetric biorthogonal wavelet bases “close” to a (nonsymmetric) orthonormal basis.