连续函数无限振荡的衰减性及其应用

本文讨论有界闭区间上无限振荡连续实函数振幅的衰减性及其在函数逼近论中的一个应用,一、证明了有界闭区间上的连续实函数若无限交错振荡(或绕曲线无限振荡),则其振幅序列必向零衰减。二、给出了连续函数在任一马尔可夫函数系所生成的线性子空间L中存在最佳一致逼近的一个充分必要条件,并以推论的形式给出了E.W.切尼的“不存在定理”。     

关于Timan定理的一点注记

本文研究了连续函数的最佳逼近多项式的点态逼近性质.通过一个具体函数的连续模估计,得到最佳逼近多项式的点态逼近阶估计,并且存在连续函数使得最佳逼近多项式能够满足Timan定理.     

关于从闭区间到完备随机赋范模的抽象值函数的Riemann 可积性的进一步研究

郭铁信和张霞最近引入和研究了从一个闭区间到一个完备随机赋范模的抽象值函数的Riemann积分, 证明了值域几乎处处有界的连续函数是Riemann 可积的. 本文首先给出该结果的一个更简短的证明, 使得我们对于值域的几乎处处有界性有一个更深的认识, 受此启发, 我们进一步构造两个例子, 其一说明值域并非几乎处处有界的连续函数也可以是Riemann 可积的, 另一例子说明连续函数可以非Riemann 可积. 最后, 我们证明从一闭区间到一个满支撑的完备随机赋范模的所有连续函数都Riemann 可积的充要条件是基底概率空间本质上由至多可数原子生成.     

微分学的几个重要概念

<正> 回顾连续函数部分,连续性是用极限定义的,间断和间断点的分类也是用极限的语言给出的.由极限的性质引进连续函数的性质.连续性作为极限过程它描述了函数在一点上变化的局部性质.在有界闭区间上连续的函数,具有有趣的总体性质.这些性质出自更深层次的极限理论.     

论广义的Канторович,Л.В.多项式及其渐近行为

§1.引言 1962年匈牙利著名数学家Freud,G.来中国讲学,作者提了一个关于非周期连续函数用线性正算子来逼近的问题,Freud,G.说:“这是他长期研究的方向。”作者解决了这个问题,研究了涉及到点x在给定闭区间上的位置的逼近,得到了Freud,G.所期待的结果。本文发展了文中的方法,研究非周期连续函数用线性正算子或线性算子来逼近,通过精巧的计算,证明了一个有趣的等式(定理2),定理2给出了函数类W~(2[3])在C空间中用线性正算子来逼近的偏差的精确值。     

一类函数列积分的性质

首先探讨了闭区间上非负连续函数列积分构成的数列极限问题,给出了极限值与函数最值有关的结论.然后利用此结论,研究了闭区间上非负连续函数列积分的第一积分中值定理"中间点"构成数列的单调性与敛散性,得到了一系列结论.     

单调连续函数两定理的证明

单调连续函数两定理的证明黄炳生（东南大学）在闭区间上连续函数的基本性质，在很多教材中都未给出证明，因而初等函数的连续性碍难懂得深透。但其中有在部分区间上是单调的。若懂得了单调连续函数性质，则收益不小。其证明则易于接受。定理１设ｆ（）在［ａ，ｂ］区间上...     

多元Stancu多项式与连续模

本文研究单纯形上多元Stancu多项式与连续模之间的关系,证明了Stancu多项式具有保持连续模的性质,推广了一元Bernstein多项式的相应结果.同时,利用多元函数的Ditzian-Totik连续模估计Stancu多项式逼近多元连续函数速度的上界和下界,得到一个使得逼近速度为O(n-a)(0相似文献   

用代数多项式逼近具有可变光滑性的函数

曾经讨论过用整函数逼近具有可变光滑性的函数,得出与Jackson定理和Bernstein定理相类似的结果. 本文讨论用代数多项式逼近具有可变光滑性的函数.设f(x)在[—1,1]上连续并且满足条件     

绝对连续函数的一个充分条件

在闭区间上,连续函数和它的差值函数若都是有限分段单调函数,则证明了该函数一定是绝对连续函数.特别地,闭区间上有限分段凸或凹的连续函数必是绝对连续函数.作为应用,给出几个绝对连续函数实例.     

Differentiability of p-Harmonic Functions on Metric Measure Spaces

We study p-harmonic functions on metric measure spaces, which are formulated as minimizers to certain energy functionals. For spaces supporting a p-Poincaré inequality, we show that such functions satisfy an infinitesmal Lipschitz condition almost everywhere. This result is essentially sharp, since there are examples of metric spaces and p-harmonic functions that fail to be locally Lipschitz continuous on them. As a consequence of our main theorem, we show that p-harmonic functions also satisfy a generalized differentiability property almost everywhere, in the sense of Cheeger’s measurable differentiable structures.     

Completeness and interpolation of almost‐everywhere quantification over finitely additive measures

We give an axiomatization of first‐order logic enriched with the almost‐everywhere quantifier over finitely additive measures. Using an adapted version of the consistency property adequate for dealing with this generalized quantifier, we show that such a logic is both strongly complete and enjoys Craig interpolation, relying on a (countable) model existence theorem. We also discuss possible extensions of these results to the almost‐everywhere quantifier over countably additive measures.     

The Average Range Characterization of the Radon–Nikodym Property

A characterization of Banach spaces possessing the Radon—Nikodym property is given in terms of the average range of additive interval functions. We prove that a Banach space X has the RNP if and only if each X-valued additive interval function possessing absolutely continuous McShane (or Henstock) variational measure has nonempty average range almost everywhere on [0, 1].     

On the convergence of a bounded amart and a conjecture of Chatterji

Through the decomposition theorem of Lebesgue and Darst it is possible to define a generalized Radon-Nikodym derivative of a bounded additive set function with respect to a bounded countably additive set function. For a bounded amart the derivatives of the components are shown to converge almost everywhere. This result, together with a characterization of amarts, yields a theorem stated by Chatterji whose proof is incorrect.     

Graphensätze für topologische Räume

In generalising a closed graph theorem of Dektjarev [5] we show that each almost closed and uniformly almost continuous relation of a quasi-uniform space to a hypercomplete quasi-uniform space is uniformly continuous. The hypothesis of being almost closed is necessary and actually weaker than the requirement that the relation considered has a closed graph. Since each topological space can be quasi-uniformized, one obtains closed graph theorems for topological spaces.As consequences we get an analogous closed graph theorem for locally uniformly almost continuous relations and a theorem concerning the completeness of the range of each continuous uniformly almost open mapping of a hypercomplete space to a uniform space.Finally we prove a closed graph theorem for relations to locally compact spaces without referring to any quasi-uniformization and thus without assumptions and statements concerning uniformity.     

A theorem in fine potential theory and applications to finely holomorphic functions

Consider the map from the fine interior of a compact set to the measures on the fine boundary given by Balayage of the unit point mass onto the fine boundary (the Keldych measure). It is shown that for any point in the domain there is a compact fine neighborhood of the point on which the map is continuous from the initial topology on the compact set to the norm topology on measures. In this paper we only prove a rather special case, the method could easily be generalized to more abstract potential spaces. One consequence of this result is a Hartog-type theorem for finely harmonic functions. We use the Hartog theorem, rational approximation theory, and results proved in a previous paper by the author to prove that the derivative of a finely holomorphic function exists everywhere and is finely holomorphic.     

A Converse to Kluvánek’s Theorem

I. Kluvánek extended the Whittaker-Kotel’nikov-Shannon (WKS) theorem to the abstract harmonic analysis setting. To do this, the ‘band limited’ condition on the spectrum of a continuous square-integrable function (analogue signal) required for classical WKS theorem is replaced by an ‘almost disjoint’ translates condition arising from the Fourier transform of the function vanishing almost everywhere outside a transversal of a compact quotient group. A converse of Kluvánek’s theorem is established, i.e., if the representation given by the abstract WKS theorem holds for a continuous square-integrable function with support of its Fourier transform essentially A, then A is a subset of a transversals of Γ/Λ     

关于开区间内连续函数最值的判定

由闭区间上连续函数的最值存在定理,讨论开区间内连续函数最值的存在性,并得出几个结论.     

Measurable rigidity of the cohomological equation for linear cocycles over hyperbolic systems

We show that any measurable solution of the cohomological equation for a Hölder linear cocycle over a hyperbolic system coincides almost everywhere with a Hölder solution. More generally, we show that every measurable invariant conformal structure for a Hölder linear cocycle over a hyperbolic system coincides almost everywhere with a continuous invariant conformal structure. We also use the main theorem to show that a linear cocycle is conformal if none of its iterates preserve a measurable family of proper subspaces of R^d. We use this to characterize closed negatively curved Riemannian manifolds of constant negative curvature by irreducibility of the action of the geodesic flow on the unstable bundle.     

Kronecker type theorems,normality and continuity of the multivariate Padé operator

Summary. For univariate functions the Kronecker theorem, stating the equivalence between the existence of an infinite block in the table of Padé approximants and the approximated function being rational, is well-known. In [Lubi88] Lubinsky proved that if is not rational, then its Padé table is normal almost everywhere: for an at most countable set of points the Taylor series expansion of is such that it generates a non-normal Padé table. This implies that the Padé operator is an almost always continuous operator because it is continuous when computing a normal Padé approximant [Wuyt81]. In this paper we generalize the above results to the case of multivariate Padé approximation. We distinguish between two different approaches for the definition of multivariate Padé approximants: the general order one introduced in [Levi76, CuVe84] and the so-called homogeneous one discussed in [Cuyt84]. Received December 19, 1994