关于Weierstrass定理的证明

对于Weierstrass聚点定理．部分教材通常多采用二分法加区间套定理进行证明．也有教材通过在有界数列中寻找一个单调子列来证明Weierstrass聚点定理,可惜证明过程存在问题．通过对证明过程中出现的问题进行分析．可得到基于单调有界定理的Weierstrass定理的证明．     

关于随机函数的Weierstrass逼近定理

定义了随机系数多项式的概念之后,将函数的Weierstrass逼近定理推广到了随机函数上.     

多元КАНТОРОВИЧ多项式的逼近定理

本文利用逼近转化原理,建立了多元多项式逼近的量化Korovkin型定理。改进和完善了[2]中的结果。     

外尔斯特拉斯逼近定理的概率证明

在外尔斯特拉斯逼近定理的各种证明中,伯恩斯坦的证明比较最普通,颇有感染力。因为它的证明是构造性的。其结论是:如果f是区间[0,1]上的连续实值函数,伯恩斯坦多项式序列     

内函数逼近定理及上溢原理的推广及应用

在κ-饱和的非标准模型中,首先推广了内函数逼近定理,并用这一推广定理证明了著名的A sco li定理;其次将上(下)溢原理推广到一般的定向集上,并证明了拓扑空间中单子的一些性质,给出了无穷小延伸定理的一个简单证明.     

关于一个次线性逼近定理的新证法

Ｄｅｍｙａｎｏｖ和Ｒｕｂｉｎｏｖ在［１］中给出了一个次线性逼近定理．本文对该定理的证明进行了修正．     

无穷维Banach空间上的逼近定理和弱内向映象的不动点定理

本文在适当的边界条件下证明了ｋｙＦａｎ［１，Ｔｈ，２］逼近定理对严格集压缩映象和凝聚映象在无穷维Ｂａｎａｃｈ空间中的圆环上也成立，作为应用，得到了弱内向映象的非零不动点定理。     

双连续n次积分C余弦函数的逼近定理

基于双连续半群概念,引入一致双连续半群序列概念,借助Laplace变换和Trotter-Kato定理,考察双连续n次积分C余弦函数与C-预解式之间的关系,得到逼近定理的稳定性条件,进而得出双连续n次积分C余弦函数逼近定理.从而对Banach空间强连续半群逼近定理和双连续半群逼近定理进行了推广,为相应抽象的Cauchy问题提供了解决方案.     

线性算子σ（L^∞,L^1）逼近的Korovkin定理

本文给出并证明了拓扑σ（Ｌ＾∞，Ｌ＾１）下线性算子逼近Ｌ＾∞函数的Ｋｏｒｏｖｋｉｎ定理。     

关于Poinare‘度量与Bers逼近定理

本文首先给出区域Ｄ的Ｐｏｉｎｃａｒｅ度量λ（ｚ）的几个有关性质，然后推广Ｂｅｒｓ逼近定理，得到主要结果如下：设Ｄ是连通数为有穷的有界区域，记Ａｑ（Ｄ）为Ｄ内满足‖ψ‖＝∫Ｄ「λ（ｚ）」＾２－│ψ（ｚ）‖ｄｘ∧ｄｚ＾－│〈∞的解件函烽ψ之全体构成的Ｂａｎａｃｈ空间，Ｒｑ（Ｄ，Ｔ）（Ｔ包含（Ｃ－Ｄ））表示Ａｑ（Ｄ）中极点在Ｔ的有理函数子空间，当Ｔ满足Ｂｅｒｓ逼近定理条件时，Ｒｑ（Ｄ，Ｔ）在Ａｑ（Ｄ）ｋ     

On the Weierstrass Preparation Theorem

An analytic function of several variables is considered. It is assumed that the function vanishes at some point. According to the Weierstrass preparation theorem, in the neighborhood of this point the function can be represented as a product of a nonvanishing analytic function and a polynomial in one of the variables. The coefficients of the polynomial are analytic functions of the remaining variables. In this paper we construct a method for finding the nonvanishing function and the coefficients of the polynomial in the form of Taylor series whose coefficients are found from an explicit recursive procedure using the derivatives of the initial function. As an application, an explicit formula describing a bifurcation diagram locally up to second-order terms is derived for the case of a double root.     

On a Generalization of the Stone–Weierstrass Theorem

A categorical version of the famous theorem of Stone and Weierstrass is formulated and studied in detail. Several applications and examples are given.     

Weierstrass theorem for monotonically semicontinuous functions

The concept of monotone semicontinuity is introduced. It is shown that every monotonically semicontinuous function with values in a space equipped with arbitrary preference relation achieves its extremes on compacts     

Some connections of monotonicity with Weierstrass extreme value theorem

The main scope of this article is to prove that if a continuous function attains its minimum before maximum on every compact interval, than it is nondecreasing. Finally, other similar results are established.     

Weierstrass functions with random phases

Consider the function

where 1$">, , and is a non-constant 1-periodic Lipschitz function. The phases are chosen independently with respect to the uniform probability measure on . We prove that with probability one, we can choose a sequence of scales such that for every interval of length , the oscillation of satisfies . Moreover, the inequality is almost surely true at every scale. When is a transcendental number, these results can be improved: the minoration is true for every choice of the phases and at every scale.

     

Weierstrass functions in Zygmund's class

Consider the function

where 1$"> and is an almost periodic function. It is well known that the function lives in the so-called Zygmund class. We prove that is generically nowhere differentiable. This is the case in particular if the elementary condition is satisfied. We also give a sufficient condition on the Fourier coefficients of which ensures that is nowhere differentiable.

     

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.     

插补空间的逼近外推定理

本文建立了拟模Abelian群上双参数算子族逼近的外推定理,所得的结果包含了DeVoreR．等人对正规逼近族之最佳逼近所建立的外推定理,且所需的条件更弱．同时从本文的结果立即可以建立起算子逼近的外推定理．     

Using simple harmonic motion to help in the search for tautochrone curves

The analysis of tautochrone problems involves the solution of integral equations. The paper shows how a reasonable assumption, based on experience with simple harmonic motion, allows one to greatly simplify such problems. Proposed solutions involve only mathematics available to students from first year calculus.