Totally positive functions through nonstationary subdivision schemes

In this paper a new class of nonstationary subdivision schemes is proposed to construct functions having all the main properties of B-splines, namely compact support, central symmetry and total positivity. We show that the constructed nonstationary subdivision schemes are asympotically equivalent to the stationary subdivision scheme associated with a B-spline of suitable degree, but the resulting limit function has smaller support than the B-spline although keeping its regularity.     

A Construction of Multiresolution Analysis on Interval

We present a concrete method of constructing multiresolution analysis on interval. The method generalizes the corresponding results of Cohen, Daubechies and Vial [Appl. Comput. Harmonic Anal., 1（1993）, 54-81]. By the use of the subdivision operator, the expressions of the constructed functions are more compact. Furthermore, the method reveals more clearly some properties of multiresolution analysis with certain approximation order.     

Scalar and Hermite subdivision schemes

A criterion of convergence for stationary nonuniform subdivision schemes is provided. For periodic subdivision schemes, this criterion is optimal and can be applied to Hermite subdivision schemes which are not necessarily interpolatory. For the Merrien family of Hermite subdivision schemes which involve two parameters, we are able to describe explicitly the values of the parameters for which the Hermite subdivision scheme is convergent.     

Estimating error bounds for quaternary subdivision schemes

This paper deals with the error estimation techniques of quaternary subdivision schemes. The estimation is expressed in terms of initial control point sequences and constants. It is independent of subdivision process and parametrization therefore its evaluation is straightforward.     

Interpolatory subdivision schemes with infinite masks originated from splines

A generic technique for the construction of diversity of interpolatory subdivision schemes on the base of polynomial and discrete splines is presented in the paper. The devised schemes have rational symbols and infinite masks but they are competitive (regularity, speed of convergence, computational complexity) with the schemes that have finite masks. We prove exponential decay of basic limit functions of the schemes with rational symbols and establish conditions, which guaranty the convergence of such schemes on initial data of power growth. Mathematics subject classifications (2000) 65D17, 65D07, 93E11     

Multiresolution Analysis by Spherical Up Functions

A new class of locally supported radial basis functions on the (unit) sphere is introduced by forming an infinite number of convolutions of "isotropic finite elements." The resulting up functions show useful properties: They are locally supported and are infinitely often differentiable. The main properties of these kernels are studied in detail. In particular, the development of a multiresolution analysis is given based on locally supported zonal functions within the reference space of square-integrable functions over the sphere.     

Families of univariate and bivariate subdivision schemes originated from quartic B-spline

Families of parameter dependent univariate and bivariate subdivision schemes are presented in this paper. These families are new variants of the Lane-Riesenfeld algorithm. So the subdivision algorithms consist of both refining and smoothing steps. In refining step, we use the quartic B-spline based subdivision schemes. In smoothing step, we average the adjacent points. The bivariate schemes are the non-tensor product version of our univariate schemes. Moreover, for odd and even number of smoothing steps, we get the primal and dual schemes respectively. Higher regularity of the schemes can be achieved by increasing the number of smoothing steps. These schemes can be nicely generalized to contain local shape parameters that allow the user to adjust locally the shape of the limit curve/surface.     

Finiteness conjecture and subdivision

The finiteness conjecture by J.C. Lagarias and Y. Wang states that the joint spectral radius of a finite set of square matrices is attained on some finite product of such matrices. This conjecture is known to be false in general. Nevertheless, we show that this conjecture is true for a big class of finite sets of square matrices used for the smoothness analysis of scalar univariate subdivision schemes with finite masks.     

Multiresolution analysis on local fields

In this paper, we first establish the fundamental framework of multiresolution on local field in wavelet analysis. Accurately, orthonormal systems consisting of integer translations of a single function φ∈L²(K) are characterized (Theorems 1 and 2), the concept of multiresolution analysis on local field (Definition 3) and construction of corresponding wavelet vectors (Theorem 3) are given. The definition of integral periodicity of a function on local field (Definition 4) is also given. An example is finally presented.     

Convergence of subdivision schemes and smoothness of limit functions

Starting with vector λ=(λ(k))_k∈Z∈?p(Z), the subdivision scheme generates a sequence of vectors by the subdivision operator

     

具有多项式衰减面具的向量细分格式

具有多项式衰减面具的向量细分方程在刻画小波Riesz基和双正交小波等方面有着重要作用.本文主要研究这类方程解的性质.向量的细分方程具有形式：Ф=∑α∈Zsa（α）（2·-α）,其中Ф=（Ф1,...,Фr）T是定义在Rs上的向量函数,a：=（a（α））α∈Zs是一个具有多项式衰减的r×r矩阵序列称为面具.关于面具a定义一个作用在（Lp（Rs））r上的线性算子Qa,Qaf：=∑α∈Zsa（α）f（2·α）.迭代格式（Qanf）n=1,2,...称为向量细分格式或向量细分算法.本文证明如果具有多项式衰减面具的向量细分格式在（L2（Rs））r中收敛,那么其收敛的极限函数将自动具有多项式衰减.另外,给出了当迭代的初始函数满足一定的条件时的向量细分格式的收敛阶.     

Error bounds for a class of subdivision schemes based on the two-scale refinement equation

Subdivision schemes are iterative procedures for constructing curves and constitute fundamental tools in computer aided design. Starting with an initial control polygon, a subdivision scheme refines the values computed in the previous step according to some basic rules. The scheme is said to be convergent if there exists a limit curve. The computed values define a control polygon in each step. This paper is devoted to estimating error bounds between the limit curve and the control polygon defined after k subdivision stages. In particular, a stop criterion of convergence is obtained. The refinement rules considered in the paper are widely used in practice and are associated with the well known two-scale refinement equation including as particular examples the schemes based on Daubechies’ filters. Our results generalize the previous analysis presented by Mustafa et al. in [G. Mustafa, F. Chen, J. Deng, Estimating error bounds for binary subdivision curves/surfaces, J. Comput. Appl. Math. 193 (2006) 596-613] and [G. Mustafa and M.S. Hashmi Subdivision depth computation for n-ary subdivision curves/surfaces, Vis. Comput. 26 (6-8) (2010) 841-851].     

Analysis of subdivision schemes for nets of functions by proximity and controllability

In this paper we develop tools for the analysis of net subdivision schemes, schemes which recursively refine nets of bivariate continuous functions defined on grids of lines, and generate denser and denser nets. Sufficient conditions for the convergence of such a sequence of refined nets, and for the smoothness of the limit function, are derived in terms of proximity to a bivariate linear subdivision scheme refining points, under conditions controlling some aspects of the univariate functions of the generated nets. Approximation orders of net subdivision schemes, which are in proximity with positive schemes refining points are also derived. The paper concludes with the construction of a family of blending spline-type net subdivision schemes, and with their analysis by the tools presented in the paper. This family is a new example of net subdivision schemes generating C¹ limits with approximation order 2.     

Vector subdivision schemes and multiple wavelets

We consider solutions of a system of refinement equations written in the form

where the vector of functions is in and is a finitely supported sequence of matrices called the refinement mask. Associated with the mask is a linear operator defined on by . This paper is concerned with the convergence of the subdivision scheme associated with , i.e., the convergence of the sequence in the -norm.

Our main result characterizes the convergence of a subdivision scheme associated with the mask in terms of the joint spectral radius of two finite matrices derived from the mask. Along the way, properties of the joint spectral radius and its relation to the subdivision scheme are discussed. In particular, the -convergence of the subdivision scheme is characterized in terms of the spectral radius of the transition operator restricted to a certain invariant subspace. We analyze convergence of the subdivision scheme explicitly for several interesting classes of vector refinement equations.

Finally, the theory of vector subdivision schemes is used to characterize orthonormality of multiple refinable functions. This leads us to construct a class of continuous orthogonal double wavelets with symmetry.

     

Subdivision schemes inL p spaces

Subdivision schemes play an important role in computer graphics and wavelet analysis. In this paper we are mainly concerned with convergence of subdivision schemes inL _p spaces (1≤p≤∞). We characterize theL _p -convergence of a subdivision scheme in terms of thep-norm joint spectral radius of two matrices associated with the corresponding mask. We also discuss various properties of the limit function of a subdivision scheme, such as stability, linear independence, and smoothness.     

Spectra of subdivision operators

Let be a sequence of complex numbers and except for finitely many . The subdivision operator associated with is the bi-infinite matrix . This operator plays an important role in wavelet analysis and subdivision algorithms. As the adjoint it is closely related to the well-known transfer operators (also called Ruelle operator).

In this paper we show that for any , the spectrum of in is always a closed disc centered at the origin. Moreover, except for finitely many points, all the points in the open disc of the spectrum lie in the residual spectrum.

     

Nonstandard finite difference schemes for reaction‐diffusion equations

We construct finite difference schemes for a particular class of one‐space dimension, nonlinear reaction‐diffusion PDEs. The use of nonstandard finite difference methods and the imposition of a positivity condition constrain the schemes to be explicit and allow the determination of functional relations between the space and time step‐sizes. The general procedure is illustrated by applying it to several important model systems of PDEs © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 201–214, 1999     

Multiresolution schemes for strongly degenerate parabolic equations in one space dimension

An adaptive finite volume method for one‐dimensional strongly degenerate parabolic equations is presented. Using an explicit conservative numerical scheme with a third‐order Runge‐Kutta method for the time discretization, a third‐order ENO interpolation for the convective term, and adding a conservative discretization for the diffusive term, we apply the multiresolution method combining two fundamental concepts: the switch between central interpolation or exact computing of numerical flux and a thresholded wavelet transform applied to cell averages of the solution to control the switch. Applications to mathematical models of sedimentation‐consolidation processes and traffic flow with driver reaction, which involve different types of boundary conditions, illustrate the computational efficiency of the new method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

On the regularity analysis of interpolatory Hermite subdivision schemes

It is well known that the critical Hölder regularity of a subdivision schemes can typically be expressed in terms of the joint-spectral radius (JSR) of two operators restricted to a common finite-dimensional invariant subspace. In this article, we investigate interpolatory Hermite subdivision schemes in dimension one and specifically those with optimal accuracy orders. The latter include as special cases the well-known Lagrange interpolatory subdivision schemes by Deslauriers and Dubuc. We first show how to express the critical Hölder regularity of such a scheme in terms of the joint-spectral radius of a matrix pair {F₀,F₁} given in a very explicit form. While the so-called finiteness conjecture for JSR is known to be not true in general, we conjecture that for such matrix pairs arising from Hermite interpolatory schemes of optimal accuracy orders a “strong finiteness conjecture” holds: ρ(F₀,F₁)=ρ(F₀)=ρ(F₁). We prove that this conjecture is a consequence of another conjectured property of Hermite interpolatory schemes which, in turn, is connected to a kind of positivity property of matrix polynomials. We also prove these conjectures in certain new cases using both time and frequency domain arguments; our study here strongly suggests the existence of a notion of “positive definiteness” for non-Hermitian matrices.     

Radial Multiresolution in Dimension Three

We present a construction of a wavelet-type orthonormal basis for the space of radial $L^2$-functions in {\bf R}$^3$ via the concept of a radial multiresolution analysis. The elements of the basis are obtained from a single radial wavelet by usual dilations and generalized translations. Hereby the generalized translation reveals the group convolution of radial functions in {\bf R}$^3$. We provide a simple way to construct a radial scaling function and a radial wavelet from an even classical scaling function on {\bf R}. Furthermore, decomposition and reconstruction algorithms are formulated.