Direct Approximation Theorem on a Family of Two Segments

Polynomial approximations are studied for some function classes. Bibliography: 6 titles.     

Direct Theorem about Approximation on a Family of Two Segments

Introduce the notation: , is the union of two segments [-1,1] and [-1 + ,1+ ], is a noninteger number, is the Hölder class with exponent on The following result announced by the authors in [J. Math. Sci. 117 (2003), No. 3] is proved. There exist numbers a ₁ ( ) , b ₁ ( ) 0 depending only on such that for any there exists a polynomial , such that . Bibliography: 11 titles.     

无穷级数中Dini定理的逆问题及其推广

本文从严格的数学分析的思想方法,研究了无穷级数中的狄尼定理,指出它的逆命题是不成立的,同时也给出了反例.后半部分在引用一个引理的基础上,证明了正项级数∑∞n=1an/rαnrβn-1也收敛,即将狄尼定理适当进行推广.     

Bernstein Inequality and Inverse Approximation Theorem in Modular Function Spaces

A Bernstein inequality and an inverse approximation theorem are proved in some classes of modular function spaces.     

球面Jackson多项式逼近的正逆定理

研究了球面Jackson多项式J_(v,s)f的逼近阶,建立了该多项式逼近的强型正向与逆向不等式.利用球面光滑模较好地刻画了Jackson多项式的逼近性能,证明了存在与v和f无关的常数C_1和C_2,使得对于定义在球面上任意p-幂勒贝格可积或连续函数f成立C_1ω(f,1/v)_p≤‖J_(v,s)f-f‖_p≤C_2ω(f,1/v)_p,其中ω(f,t)p是f的光滑模.     

球面Jackson多项式逼近的正逆定理

研究了球而Jackson多项式Jv,sf的逼近阶,建立立该多项式逼近的强型正向与逆向不等式.利用球面光滑模较好地刻画了Jackson多项式的逼近性能,证明了存在与v和f无关的常数C1和C2,使得对于定义在球面上任意p-幂勒贝格可积或连续函数f成立C1ω(f,1/v)p≤‖Jv,sf-f‖p≤C2ω(f,1/v)p,其中ω(f,t)p是f的光滑模.     

Inverse Theorem of Approximation in L_p for the Stancu Integral-type Operators on a Simplex

In this paper, the integral-type Stancu operators on a simplex is considered and its inverse theorem of approximation in Lp(1≤p< ∞)has been obtained.     

单纯形上积分型Stancu算子在Lp中的逼近逆定理

In this paper, the integral-type Stancu operators on a simplex is considered and its inverse theorem of approximation in Lp(1≤p<+∞)has been obtained.     

Approximation by Entire Functions on a Countable Union of Segments on the Real Axis: 3. Further Generalization

In this paper, an approximation of functions of extensive classes set on a countable unit of segments of a real axis using the entire functions of exponential type is considered. The higher the type of the approximating function is, the higher the rate of approximation near segment ends can be made, compared with their inner points. The general approximation scale, which is nonuniform over its segments, depending on the type of the entire function, is similar to the scale set out for the first time in the study of the approximation of the function by polynomials. For cases with one segment and its approximation by polynomials, this scale has allowed us to connect the so-called direct theorems, which state a possible rate of smooth function approximation by polynomials, and the inverse theorems, which give the smoothness of a function approximated by polynomials at a given rate. The approximations by entire functions on a countable unit of segments for the case of Hölder spaces have been studied by the authors in two preceding papers. This paper significantly expands the class of spaces for the functions, which are used to plot an approximation that engages the entire functions with the required properties.     

Existence Theorem in the Optimal Control Problem on an Infinite Time Interval

We consider the optimal control problem on an infinite time interval. The system is linear in the control, the functional is convex in the control, and the control set is convex and compact. We propose a new condition on the behavior of the functional at infinity, which is weaker than the previously known conditions, and prove the existence theorem for the solution under this condition. We consider several special cases and propose a general abstract scheme.     

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.     

Approximation of an Inverse Initial Problem for a Biparabolic Equation

In this paper, we consider the problem of finding the initial distribution for the linear inhomogeneous and nonlinear biparabolic equation. The problem is severely ill-posed in the sense of Hadamard. First, we apply a general filter method to regularize the linear nonhomogeneous problem. Then, we also give a regularized solution and consider the convergence between the regularized solution and the sought solution. Under the a priori assumption on the exact solution belonging to a Gevrey space, we consider a generalized nonlinear problem by using the Fourier truncation method to obtain rigorous convergence estimates in the norms on Hilbert space and Hilbert scale space.     

Infinite Dimensional Bicomplex Spectral Decomposition Theorem

This paper presents a bicomplex version of the Spectral Decomposition Theorem on infinite dimensional bicomplex Hilbert spaces. In the process, the ideas of bounded linear operators, orthogonal complements and compact operators on bicomplex Hilbert spaces are introduced and treated in relation with the classical Hilbert space M′ imbedded in any bicomplex Hilbert space M.     

Polynomial Approximation in the L p-Metric on Disjoint Segments

The Sobolev functional class on a finite disjoint union of segments is described in terms of the rate of polynomial approximation. Bibliography: 9 titles.     

Numerical Approximation of Neutral Differential Equations on Infinite Interval

In this paper we study numerical approximation of linear neutral differential equations on infinite interval using equations with piecewise constant arguments. As an application of our approximation results, we obtain stability theorems for some classes of linear delay and neutral difference equations.     

Approximation of WKS Sampling Theorem on Random Signals

The WKS sampling theorem is a fundamental result in the field of telecommunication and signal processing. It is a perfect example of the synthesis of traditionally distinct disciplines in mathematics, engineering analysis, and the sciences. The theorem recovers a bandlimited signal from its samples at uniformly points, which purpose is to analyze deterministic functions, or deterministic signals. However, there are a few counterparts of this theorem for random signals so far. This article gives a new WKS sampling principle on random signals from local averages with probability 1.     

Infinite Matroidal Version of Hall's Matching Theorem

Hall's theorem for bipartite graphs gives a necessary and sufficientcondition for the existence of a matching in a given bipartitegraph. Aharoni and Ziv have generalized the notion of matchabilityto a pair of possibly infinite matroids on the same set andgiven a condition that is sufficient for the matchability ofa given pair (M, W) of finitary matroids, where the matroidM is SCF (a sum of countably many matroids of finite rank).The condition of Aharoni and Ziv is not necessary for matchability.The paper gives a condition that is necessary for the existenceof a matching for any pair of matroids (not necessarily finitary)and is sufficient for any pair (M, W) of finitary matroids,where the matroid M is SCF.     

On the Choice of Segments in Piecewise Approximation

An optimal choice of segment boundaries in piecewise approximationis shown to be soluble by means of a dynamic programme in ascalar state variable. It is shown that the composite errorfunction is continuous in modulus, and that for approximationof concave or convex functions by linear segments, the compositeapproximation is continuous. Provided the residual term satisfiescertain requirements, the solution has uniqueness propertiesand can be found by means of a grid search method.