Smoothness Analysis of Nonlinear Subdivision Schemes of Homogeneous and Affine Invariant Type

Nonlinear subdivision schemes arise from, among other applications, nonlinear multiscale signal processing and shape preserving interpolation. For the univariate homogeneous subdivision operator $S:\ell(\bZ) \goto \ell(\bZ)$ we establish a set of commutation/recurrence relations which can be used to analyze the asymptotic decay rate of $\|\Delta^r S^j m\|_{\ell^\infty}$, $j=1,2,\ldots,$ the latter in turn determines the convergence and H\older regularity of $S$. We apply these results to prove that the critical H\older regularity exponent of a nonlinear subdivision scheme based on median-interpolation is equal to that of an approximating linear subdivision scheme, resolving a conjecture by Donoho and Yu. We also consider a family of nonlinear but affine invariant subdivision operators based on interpolation-imputation of $p$-mean (of which median corresponds to the special case $p=1$) as well as general continuous $M$-estimators. We propose a linearization principle which, when applied to $p$-mean subdivision operators, leads to a family of linear subdivision schemes. Numerical evidence indicates that in at least many cases the critical smoothness of a $p$-mean subdivision scheme is the same as that of the corresponding linear scheme. This suggests a more coherent view of the result obtained in this paper.     

Interpolatory wavelets for manifold-valued data

Geometric wavelet-like transforms for univariate and multivariate manifold-valued data can be constructed by means of nonlinear stationary subdivision rules which are intrinsic to the geometry under consideration. We show that in an appropriate vector bundle setting for a general class of interpolatory wavelet transforms, which applies to Riemannian geometry, Lie groups and other geometries, Hölder smoothness of functions is characterized by decay rates of their wavelet coefficients.     

On a continuous form of Hölder inequality

A continuous form of Hölder inequality is considered here extending some known results.     

Morrey Regularity of Nonlinear Elliptic Systems of High Order under Degeneration of Ellipticity

We study nonlinear elliptic systems of the form , even, , with the natural energy space . We establish that for solutions from belong to the Morrey space and the Morrey exponent does not tend to zero under the degeneration of ellipticity. In the case , a similar result is obtained under an additional structure condition on the system.     

Estimation of Local Smoothness Coefficients for Continuous Time Processes

We consider processes that satisfied a local Hölder condition with coefficient ₀. According to the sampling times of observations given by i _n with i=0,...,n–1, we study two general classes of estimators for ₀. Their almost sure rates of convergence depend on asymptotic independence of the observed processes, on _n and eventually on an extra parameter ₀. Since this last parameter is in general unknown, we construct a family of preliminary estimators for ₀ with their rates of almost sure convergence. Finally we present some numerical simulations in order to compare the behaviour of our various estimators.     

一类保凸的细分格式

提出了一类新的具有高阶连续性和保凸性的5-点二重细分格式.可以证明在参数的某些取值范围内,极限曲线是$C^k~(k=0,1,\ldots,7)$连续的.本文还给出了极限曲线保凸时的参数的取值范围.数值例子表明该格式是灵活有效的.     

Generalized synchronization of two bidirectionally coupled discrete dynamical systems

Based on the modified system approach the generalized synchronization (GS) in two bidirectionally coupled discrete dynamical systems is classified into several types, and under some conditions, the existence, Lipschitz smoothness and Hölder continuity of two kinds of GS therein are derived and theoretically proved. In addition, numerical simulations validate the present theory.     

The Generalized Multifractional Brownian Motion

It is well known that the fractional Brownian motion (FBM) is of great interest in modeling. However, its Hölder is the same all along its path and this restricts its field of application. Therefore, it would be useful to construct a Gaussian process extending the FBM and having a Hölder that is allowed to change. A partial answer to this problem is supplied by the multifractional Brownian motion (MBM); but the Hölder of the MBM must necessarily be continuous and this may be a drawback in some situations. In this paper we construct a Gaussian process generalizing the MBM and having a Hölder that can be a very irregular function.     

Conjugacies for linear and nonlinear perturbations of nonuniform behavior

We construct topological conjugacies between linear and nonlinear evolution operators that admit either a nonuniform exponential contraction or a nonuniform exponential dichotomy. We consider evolution operators defined by nonautonomous differential equations x^′=A(t)x+f(t,x) in a Banach space. The conjugacies are obtained by first considering sufficiently small linear and nonlinear perturbations of linear equations x^′=A(t)x. In the case of linear perturbations, we construct in a more or less explicit manner topological conjugacies between the two linear flows. In the case of nonlinear perturbations, we obtain a version of the Grobman-Hartman theorem for nonuniformly hyperbolic dynamics. Furthermore, all the conjugacies that we construct are locally Hölder continuous provided that the vectors fields are of class C¹. As a byproduct of our approach, we give conditions for the robustness of strong nonuniform exponential behavior, in the sense that under sufficiently small perturbations the structure determined by the stable and unstable bundles persists up to small variations. We also show that the constants determining the nonuniform exponential contraction or nonuniform exponential dichotomy vary continuously with the perturbation. All the results are obtained in Banach spaces.     

Smoothness of Stationary Subdivision on Irregular Meshes

We derive necessary and sufficient conditions for tangent plane and C ^k -continuity of stationary subdivision schemes near extraordinary vertices. Our criteria generalize most previously known conditions. We introduce a new approach to analysis of subdivision surfaces based on the idea of the universal surface . Any subdivision surface can be locally represented as a projection of the universal surface, which is uniquely defined by the subdivision scheme. This approach provides us with a more intuitive geometric understanding of subdivision near extraordinary vertices. February 16, 1998. Date revised: January 27, 1999. Date accepted: April 2, 1999.     

Some Further Generalizations of Ostrowski Inequality for Hölder Functions and Functions with Bounded Derivatives

Some generalizations of Ostrowski inequality for Hölder functions and functions with bounded derivatives are given.     

Regularity for a class of degenerate elliptic equations with discontinuous coefficients under natural growth

In this paper we are concerned with the regularity in Morrey spaces for weak solutions of a class of degenerate elliptic equations when the coefficient matrices satisfy certain VMO conditions in x uniformly with respect to u and the lower order terms satisfy a natural growth condition. Interior Hölder continuity of weak solutions is also derived with the improvement of the given data regularities.     

Subdivision Schemes and Irregular Grids

The present paper deals with subdivision schemes associated with irregular grids. We first give a sufficient condition concerning the difference scheme to obtain convergence. This condition generalizes a necessary and sufficient condition for convergence known in the case of uniform and stationary schemes associated with a regular grid. Through this sufficient condition, convergence of a given subdivision scheme can be proved by comparison with another scheme. Indeed, when two schemes are equivalent in some sense, and when one satisfies the sufficient condition for convergence, the other also satisfies it and it therefore converges too. We also study the smoothness of the limit functions produced by a scheme which satisfies the sufficient condition. Finally, the results are applied to the study of Lagrange interpolating subdivision schemes of any degree, with respect to particular irregular grids.     

基于一种新的集合插值的非凸紧集的细分概型

对A rtstein给出的度量平均的定义作了改进,给出一种新的集合插值,并基于这种新的集合插值,对相应的关于一般紧集的样条细分和插值细分分别作了研究,并给出了细分的收敛性性质.与此同时,将这种新的集合插值与基于度量平均的插值及基于M inkow sk i平均的插值分别作了比较,可以看出新的集合插值在某些方面具有更好的物理性质.     

Schauder estimates for parabolic nondivergence operators of Hörmander type

Let X₁,X₂,…,Xq be a system of real smooth vector fields satisfying Hörmander's rank condition in a bounded domain Ω of Rn. Let be a symmetric, uniformly positive definite matrix of real functions defined in a domain U⊂R×Ω. For operators of kind

     

基于四点插值细分小波的SAR图像去噪

针对SAR图像去噪过程中存在降低相干斑与保持有效细节这一矛盾,提出了一种基于四点插值细分小波的SAR图像去噪算法,该方法将小波和细分方法相融合,将四点插值细分规则应用到细分小波中,提出了图像去噪的新方法.该算法先用四点插值细分小波对原始图像进行分解,然后用Bayes自适应阈值及阈值函数对图像进行去噪,最后对去噪的小波系...     

On Bivariate Smoothness Spaces Associated with Nonlinear Approximation

In recent years there have been various attempts at the representations of {\mbox multivariate} signals such as images, which outperform wavelets. As is well known, wavelets are not optimal in that they do not take full advantage of the geometrical regularities and singularities of the images. Thus these approaches have been based on tracing curves of singularities and applying bandlets, curvelets, ridgelets, etc., or allocating some weights to curves of singularities like the Mumford–Shah functional and its modifications. In the latter approach a function is approximated on subdomains where it is smoother but there is a penalty in the form of the total length (or other measurement) of the partitioning curves. We introduce a combined measure of smoothness of the function in several dimensions by augmenting its smoothness on subdomains by the smoothness of the partitioning curves. Also, it is known that classical smoothness spaces fail to characterize approximation spaces corresponding to multivariate piecewise polynomial nonlinear approximation. We show how the proposed notion of smoothness can almost characterize these spaces. The question whether the characterization proposed in this work can be further simplified remains open.     

Continuous Dependence Estimates for Viscosity Solutions of Fully Nonlinear Degenerate Parabolic Equations

Using the maximum principle for semicontinuous functions (Differential Integral Equations3 (1990), 1001-1014; Bull. Amer. Math. Soc. (N.S)27 (1992), 1-67), we establish a general “continuous dependence on the non- linearities” estimate for viscosity solutions of fully nonlinear degenerate parabolic equations with time- and space-dependent nonlinearities. Our result generalizes a result by Souganidis (J. Differential Equations56 (1985), 345-390) for first- order Hamilton-Jacobi equations and a recent result by Cockburn et al. (J. Differential Equations170 (2001), 180-187) for a class of degenerate parabolic second-order equations. We apply this result to a rather general class of equations and obtain: (i) Explicit continuous dependence estimates. (ii) L^∞ and Hölder regularity estimates. (iii) A rate of convergence for the vanishing viscosity method. Finally, we illustrate results (i)-(iii) on the Hamilton-Jacobi- Bellman partial differential equation associated with optimal control of a degenerate diffusion process over a finite horizon. For this equation such results are usually derived via probabilistic arguments, which we avoid entirely here.     

Convexity with respect to Hölder mean involving zero-balanced hypergeometric functions

In this note we investigate the convexity of zero-balanced hypergeometric functions with respect to Hölder mean.     

Fractional Integrals of Imaginary Order in the Space of Hölder Functions with Polynomial Weight on an Interval

We study fractional integrals of imaginary order in the Marchaud form in Hölder space with polynomial weight at finitely many points in the closed interval [0,1]. We present conditions on the orders of the weight for which this operator is invertible in the space described above.