Fourier Expansions of Piecewise Smooth Functions

Although Fourier series or integrals of piecewise smooth functions may be slowly convergent, sometimes it is possible to accelerate their speed of convergence by adding and subtracting suitable combination of known functions.     

A Property of Piecewise Smooth Functions

Piecewise smooth equations are increasingly important in the numerical treatment of complementarity problems and models of equilibrium. This note brings out a property of the functions that enter such equations, for instance through penalty expressions.     

Melnikov Functions for a Class of Piecewise Hamiltonian Systems

This paper is concerned with the number of limit cycles for a class of piecewise Hamiltonian systems with two zones separated by two semi-straight lines. By constructing a Poincar\''{e} map, we obtain explicit expressions of the first, second and third order Melnikov functions. In addition, we apply their expressions to give upper bounds of the number of limit cycles bifurcated from a period annulus of a piecewise polynomial Hamiltonian system.     

一元函数的极值及其奇异性

针对一元函数的极值理论,为学生介绍这一理论的进一步发展即奇点理论的一些基本概念和理论.指出二者之间的密切联系.通过对平面曲线的高斯映射以及奇点的介绍,向学生展示高斯曲率与函数的二阶导数以及高斯映射的奇点与函数的拐点之间的奇妙关系.并指出它们的几何意义.同时强调在学习高等数学的相关内容时,应该养成思考它们的几何背景及意义...     

Bifurcation in a Class of Planar Piecewise Smooth Systems with 3-parameters

This paper is concerned with the bifurcation properties on the line of discontinuity of planar piecewise smooth systems. The existence of equilibria and periodic solutions with sliding motion in a class of planar piecewise smooth systems with 3-parameters is investigated in this paper using the theory of differential inclu-sion and tools of Poincar′e maps.     

Convergence of Lagrange Interpolation Polynomials for Piecewise Smooth Functions

The present paper first establishes a decomposition result for f(x)∈ C ^r C ^r+1. By using this decomposition we thus can obtain an estimate of ∣f(x) - L _n (f,x)∣ which reflects the influence of the position of the x's and ω(f ^(r+1),δ)_j, j = 0,1,...,s, on the error of approximation. This revised version was published online in June 2006 with corrections to the Cover Date.     

Bifurcation of Equilibria in a Class of Planar Piecewise Smooth Systems with 3 Parameters

The goal of this paper is to investigate the bifurcation properties of stationary points of a class of planar piecewise smooth systems with 3 parameters using the theory of differential inclusions. We especially study the existence of the stationary points on the line of discontinuity of this kind of planar piecewise smooth system.     

Sets of Uniform Convergence of Fourier Expansions of Piecewise Smooth Functions

For Fourier expansions of piecewise smooth functions associated with an elliptic operator of order m in Rⁿ, the sets of uniform convergence are described.     

The Weierstrass Transform for a Class of Generalized Functions

The classical theory of the Weierstrass transform is extended to a generalized function space which is the dual of a testing function space consisting of purely entire functions with certain growth conditions developed by Kenneth B. Howell. An inversion formula and characterizations for this transform are obtained. A comparative study with the existing literature is also undertaken.     

Fourier Reconstruction of Functions from their Nonstandard Sampled Radon Transform

In this article, we suggest a new Fourier transform based algorithm for the reconstruction of functions from their nonstandard sampled Radon transform. The algorithm incorporates recently developed fast Fourier transforms for nonequispaced data. We estimate the corresponding aliasing error in dependence on the sampling geometry of the Radon transform and confirm our theoretical results by numerical examples.     

The Problem of the Pointwise Fourier Inversion for Piecewise Smooth Functions of Several Variables

We study the pointwise convergence problem for the inverse Fourier transform of piecewise smooth functions, i.e., whether S_rD f (\bx) ? f (\bx)S_{\rho D} f (\bx) \to f (\bx) as r? ￥\rho \to \infty . r? ￥\rho \to \infty . Here for \bx,\bxi ? \Rn\bx,\bxi \in \Rn S_rDf(\bmx)=\dsf1(2p)^n/2\intli_rD [^(f)](\bxi) e^{\dst iá\bmx,\bxi?} d\bxi . S_{\rho D}f(\bm{x})=\dsf1{(2\pi)^{n/2}}\intli_{\rho D} \widehat{f}(\bxi) e^{\dst i\langle\bm{x},\bxi\rangle} d\bxi~. is the partial sum operator using a convex and open set DD containing the origin, and rD={ r\bxi:\bxi ? D }\rho D=\left\{ \rho \bxi:\bxi\in D \right\}.     

Application of the Regularization Method to the Numberical Inversion of a Class of Generalized Radon Transform

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On the Eigenfunction Expansion of a Piecewise Smooth Function

For eigenfunction expansions of piecewise smooth functionsassociated with an elliptic operator of order m in R² theconvergence is proved.     

Radon Transform on the Torus

We consider the Radon transform on the (flat) torus \mathbbTⁿ = \mathbbRⁿ/\mathbbZⁿ{\mathbb{T}^{n} = \mathbb{R}^{n}/\mathbb{Z}^n} defined by integrating a function over all closed geodesics. We prove an inversion formula for this transform and we give a characterization of the image of the space of smooth functions on \mathbbTⁿ{\mathbb{T}^{n}} .     

Singularities and Limit Functions in Iteration of Meromorphic Functions

Let f(z) be a transcendental meromorphic function. The paperinvestigates, using the hyperbolic metric, the relation betweenthe forward orbit P(f) of singularities of f^–1 and limitfunctions of iterations of f in its Fatou components. It ismainly proved, among other things, that for a wandering domainU, all the limit functions of {fⁿ|_U} lie in the derived setof P(f) and that if f^np|_V q(n +) for a Fatou component V, theneither q is in the derived set of S_p (f) or f^p(q) = q. As applicationsof main theorems, some sufficient conditions of the non-existenceof wandering domains and Baker domains are given.