关于weierstrass逼近定理的几点注记

Weierstrass逼近定理是函数逼近论中的重要定理之一,定理阐述了闭区间上的连续函数可以用一多项式去逼近.将该定理进行推广:即使一个函数是几乎处处连续的,也不一定具有与连续函数相类似的逼近性质,但是一个处处不连续的函数却有可能具有这样的性质.证明了定义在闭区间上且与连续函数几乎处处相等的函数具有类似的逼近性质,并给出了weierstrass逼近定理的一个推广应用.     

Polynomial function and derivative approximation of Sinc data

Sinc methods consist of a family of one dimensional approximation procedures for approximating nearly every operation of calculus. These approximation procedures are obtainable via operations on Sinc interpolation formulas. Nearly all of these approximations–except that of differentiation–yield exceptional accuracy. The exception: when differentiating a Sinc interpolation formula that gives an approximation over an interval with a finite end-point. In such cases, we obtain poor accuracy in the neighborhood of the finite end-point. In this paper we derive novel polynomial-like procedures for differentiating a function that is known at Sinc points, to obtain an approximation of the derivative of the function that is uniformly accurate on the whole interval, finite or infinite, in the case when the function itself has a derivative on the closed interval.     

Fractal Polynomials and Maps in Approximation of Continuous Functions

The primary goal of this article is to establish some approximation properties of fractal functions. More specifically, we establish that a monotone continuous real-valued function can be uniformly approximated with a monotone fractal polynomial, which in addition agrees with the function on an arbitrarily given finite set of points. Furthermore, the simultaneous approximation and \mboxinterpolation which is norm-preserving property of fractal polynomials is established. In the final part of the article, we establish differentiability of a more general class of fractal functions. It is shown that these smooth fractal functions and their derivatives are good approximants for the original function and its \mboxderivatives.     

A Noise-Shaping Theorem

Noise shaping is a process which aims to remove as much quantization noise as possible from the spectrum of a given band-limited signal when quantizing it (recall that the spectrum of a signal is the support of its Fourier transform). We provide a mathematical analysis of such a process, using methods of harmonic analysis on the unit disc. We are especially interested in conditions under which an analytic (but not necessary polynomial) function may be used as a transfer function of the process. Stability conditions will be given under terms of metric characteristics of the transfer function (stability means that the quantizer is never overloaded). Several kinds of quantizer transfer functions will be considered, especially midtread and midriser ones. We shall restrict our analysis to the case of deterministic (i.e., not random) signals. Some knowledge of Fourier analysis and signal processing will be presupposed.     

Representing Superoscillations and Narrow Gaussians with Elementary Functions

A simple addition to the collection of superoscillatory functions is constructed, in the form of a square-integrable sinc function which is band-limited yet in some intervals oscillates faster than its highest Fourier component. Two parameters enable tuning of the local frequency of the superoscillations and the length of the interval over which they occur. Away from the superoscillatory intervals, the function rises to exponentially large values. An integral transform generates other band-limited functions with arbitrarily narrow peaks that are locally Gaussian. In the (delicate) limit of zero width, these would be Dirac delta-functions, which by superposition could enable construction of band-limited functions with arbitrarily fine structure.     

The Whittaker–Kotel’nikov– Shannon Theorem,Spectral Translates and Plancherel’s Formula

The classical sampling or WKS theorem on reconstructing signals from uniformly spaced samples assumes the signals to be band-limited (i.e., with spectrum in a bounded interval [-W,W]). This assumption was later weakened to a disjoint translates condition on the spectrum which led to an extension of the sampling theorem to multi-band signals with spectra in a union of finite intervals. In this article the disjoint translates condition is replaced by a more natural null intersection condition on spectral translates. This condition is shown to be equivalent to an analogue of Plancherels isometric formula when the spectrum has finite measure. Thus to some extent multi-band sampling theory has a logical structure similar to classical Fourier analysis. The relationship between the null intersection condition and the isometric formula is illustrated by considering the consequences when the null intersection condition does not hold. In this case, the sampling representation cannot hold for any function, whereas the isometric formula can still hold for some functions.     

The varying piecewise interpolation solution of the Cauchy problem for ordinary differential equations with iterative refinement

A piecewise interpolation approximation of the solution to the Cauchy problem for ordinary differential equations (ODEs) is constructed on a set of nonoverlapping subintervals that cover the interval on which the solution is sought. On each interval, the function on the right-hand side is approximated by a Newton interpolation polynomial represented by an algebraic polynomial with numerical coefficients. The antiderivative of this polynomial is used to approximate the solution, which is then refined by analogy with the Picard successive approximations. Variations of the degree of the polynomials, the number of intervals in the covering set, and the number of iteration steps provide a relatively high accuracy of solving nonstiff and stiff problems. The resulting approximation is continuous, continuously differentiable, and uniformly converges to the solution as the number of intervals in the covering set increases. The derivative of the solution is also uniformly approximated. The convergence rate and the computational complexity are estimated, and numerical experiments are described. The proposed method is extended for the two-point Cauchy problem with given exact values at the endpoints of the interval.     

Characterization of local sampling sequences for spline subspaces

The sampling theorem is one of the most powerful tools in signal analysis. It says that to recover a function in certain function spaces, it suffices to know the values of the function on a sequence of points. Most of known results, e.g., regular and irregular sampling theorems for band-limited functions, concern global sampling. That is, to recover a function at a point or on an interval, we have to know all the samples which are usually infinitely many. On the other hand, local sampling, which invokes only finite samples to reconstruct a function on a bounded interval, is practically useful since we need only to consider a function on a bounded interval in many cases and computers can process only finite samples. In this paper, we give a characterization of local sampling sequences for spline subspaces, which is equivalent to the celebrated Schönberg-Whitney Theorem and is easy to verify. As applications, we give several local sampling theorems on spline subspaces, which generalize and improve some known results.     

Generalized solution in singular stochastic control: The nondegenerate problem

This paper is concerned with singular stochastic control for non-degenerate problems. It generalizes the previous work in that the model equation is nonlinear and the cost function need not be convex. The associated dynamic programming equation takes the form of variational inequalities. By combining the principle of dynamic programming and the method of penalization, we show that the value function is characterized as a unique generalized (Sobolev) solution which satisfies the dynamic programming variational inequality in the almost everywhere sense. The approximation for our singular control problem is given in terms of a family of penalized control problems. As a result of such a penalization, we obtain that the value function is also the minimum cost available when only the admissible pairs with uniformly Lipschitz controls are admitted in our cost criterion.     

Decomposition theorems and approximation by a ``floating" system of exponentials

The main problem considered in this paper is the approximation of a trigonometric polynomial by a trigonometric polynomial with a prescribed number of harmonics. The method proposed here gives an opportunity to consider approximation in different spaces, among them the space of continuous functions, the space of functions with uniformly convergent Fourier series, and the space of continuous analytic functions. Applications are given to approximation of the Sobolev classes by trigonometric polynomials with prescribed number of harmonics, and to the widths of the Sobolev classes. This work supplements investigations by Maiorov, Makovoz and the author where similar results were given in the integral metric.

     

The polynomial Fourier transform with minimum mean square error for noisy data

In 2006, Naoki Saito proposed a Polyharmonic Local Fourier Transform (PHLFT) to decompose a signal f∈L²(Ω) into the sum of a polyharmonic componentu and a residualv, where Ω is a bounded and open domain in R^d. The solution presented in PHLFT in general does not have an error with minimal energy. In resolving this issue, we propose the least squares approximant to a given signal in L²([−1,1]) using the combination of a set of algebraic polynomials and a set of trigonometric polynomials. The maximum degree of the algebraic polynomials is chosen to be small and fixed. We show in this paper that the least squares approximant converges uniformly for a Hölder continuous function. Therefore Gibbs phenomenon will not occur around the boundary for such a function. We also show that the PHLFT converges uniformly and is a near least squares approximation in the sense that it is arbitrarily close to the least squares approximant in L² norm as the dimension of the approximation space increases. Our experiments show that the proposed method is robust in approximating a highly oscillating signal. Even when the signal is corrupted by noise, the method is still robust. The experiments also reveal that an optimum degree of the trigonometric polynomial is needed in order to attain the minimal l² error of the approximation when there is noise present in the data set. This optimum degree is shown to be determined by the intrinsic frequency of the signal. We also discuss the energy compaction of the solution vector and give an explanation to it.     

Approximation for spherical functions by Bochner-Riesz means

This paper deals with approximation of functions on the unit sphere. The uniform and almost everywhere approximation properties of Bochner-Riesz means at and below the critical index for spherical harmonic expansions are studied. The results obtained here are analogous to those of multiple Fourier series. Supported by NSFC     

ON THE COUPLING OF BOUNDARY INTEGRAL AND FINITE ELEMENT METHODS FOR SIGNORINI PROBLEMS

1.IntroductionPartialdifferentialequationssubjecttounilateralboundaryconditionsareusuallycalledSignoriniproblemsintheliterature.TheseproblemshavebeenstudiedbymanyauthodssincetheappearenceofthehistoricalpaperbyA.Signoriniin1933[25].Signoriniproblemsaroseinmanyareasofapplicationse.g.,theelasticitywithunilateralconditions[lo],thefluidmechnicsproblemsinmediawithsemipermeableboundaries[8,12],theelectropaintprocess[1]etc.Fortheexistence,uniquenessandregularityresultsforSignorinitypeproblemswerefer…     

Numerical Projection Method For Inverse Fourier Transform And Its Application

Numerical projection method of the Fourier transform inversion from data given on a finite interval is proposed. It is based on an expansion of the solution into a series of eigenfunctions of the Fourier transform. The number of terms of the expansion depends on the length of the data interval. Convergence of the solution of the method is proved. The projection method for the case of the sine Fourier transform and the set of the odd Hermite functions being its eigenfunctions are examined and applied to numerical Fourier filtering.     

Approximating Probability Distributions Using Small Sample Spaces

We formulate the notion of a "good approximation" to a probability distribution over a finite abelian group ?. The quality of the approximating distribution is characterized by a parameter ɛ which is a bound on the difference between corresponding Fourier coefficients of the two distributions. It is also required that the sample space of the approximating distribution be of size polynomial in and 1/ɛ. Such approximations are useful in reducing or eliminating the use of randomness in certain randomized algorithms. We demonstrate the existence of such good approximations to arbitrary distributions. In the case of n random variables distributed uniformly and independently over the range , we provide an efficient construction of a good approximation. The approximation constructed has the property that any linear combination of the random variables (modulo d) has essentially the same behavior under the approximating distribution as it does under the uniform distribution over . Our analysis is based on Weil's character sum estimates. We apply this result to the construction of a non-binary linear code where the alphabet symbols appear almost uniformly in each non-zero code-word. Received: September 22, 1990/Revised: First revision November 11, 1990; last revision November 10, 1997     

一类双曲型方程的Hamilton算子半群方法

将一类双曲型方程混合问题转换成一阶抽象Cauchy问题,证明所得Hamilton算子矩阵H在相应空间中生成压缩半群,并借助Fourier变换,采用一致连续半群做逼近的方法,得到H所生成的压缩半群,进而给出了问题的古典解.     

Fourier Series with Linear Splines on an Interval

We construct orthonormal bases of linear splines on a finite interval [a, b] and then we study the Fourier series associated to these orthonormal bases. For continuous functions defined on [a, b], we prove that the associated Fourier series converges pointwisely on (a, b) and also uniformly on [a, b], if it convergences pointwisely at a and b.     

Determination of periodic orbits of nonlinear oscillators by means of piecewise-linearization methods

An approximate method based on piecewise linearization is developed for the determination of periodic orbits of nonlinear oscillators. The method is based on Taylor series expansions, provides piecewise analytical solutions in three-point intervals which are continuous everywhere and explicit three-point difference equations which are P-stable and have an infinite interval of periodicity. It is shown that the method presented here reduces to the well-known Störmer technique, is second-order accurate, and yields, upon applying Taylor series expansion and a Padé approximation, another P-stable technique whenever the Jacobian is different from zero. The method is generalized for single degree-of-freedom problems that contain the velocity, and (approximate) analytical solutions are presented. Finally, by introducing the inverse of a vector and the vector product and quotient, and using Taylor series expansions and a Padé approximation, the method has been generalized to multiple degree-of-freedom problems and results in explicit three-point finite difference equations which only involve vector multiplications.     

An almost third order finite difference scheme for singularly perturbed reaction-diffusion systems

This paper addresses the numerical approximation of solutions to coupled systems of singularly perturbed reaction-diffusion problems. In particular a hybrid finite difference scheme of HODIE type is constructed on a piecewise uniform Shishkin mesh. It is proved that the numerical scheme satisfies a discrete maximum principle and also that it is third order (except for a logarithmic factor) uniformly convergent, even for the case in which the diffusion parameter associated with each equation of the system has a different order of magnitude. Numerical examples supporting the theory are given.     

Uncertainty principles for orthonormal sequences

The aim of this paper is to provide complementary quantitative extensions of two results of H.S. Shapiro on the time-frequency concentration of orthonormal sequences in L²(R). More precisely, Shapiro proved that if the elements of an orthonormal sequence and their Fourier transforms are all pointwise bounded by a fixed function in L²(R) then the sequence is finite. In a related result, Shapiro also proved that if the elements of an orthonormal sequence and their Fourier transforms have uniformly bounded means and dispersions then the sequence is finite. This paper gives quantitative bounds on the size of the finite orthonormal sequences in Shapiro's uncertainty principles. The bounds are obtained by using prolate spheroïdal wave functions and combinatorial estimates on the number of elements in a spherical code. Extensions for Riesz bases and different measures of time-frequency concentration are also given.