Uniform continuity on unbounded intervals

We present a teaching approach to uniform continuity on unbounded intervals which, hopefully, may help to meet the following pedagogical objectives:

1. To provide students with efficient and simple criteria to decide whether a continuous function is also uniformly continuous;

2. To provide students with skill to recognize graphically significant classes of both uniformly and nonuniformly continuous functions.

Assembling some well-known facts and refining the resulting statement, we establish a useful asymptotic coincidence test for the uniform continuity on unbounded intervals. That test is the core of the present note and yields an easily applicable technique. In particular, one of its immediate consequences is the elementary fact that continuity and existence of horizontal or oblique asymptotes imply uniform continuity.     

On the estimate of the uniform deviation from the class of functions with bounded third derivative

The problem of the uniform approximation of a continuous function on a closed interval by a class of functions with a uniformly bounded third derivative is considered. It is shown that the value of best approximation of a function by this class cannot be estimated linearly in terms of its third-order modulus of continuity. At the same time, such estimates exist for classes with bounded first or second derivatives.     

On the estimate of the uniform deviation from the class of functions with bounded third derivative

The problem of the uniform approximation of a continuous function on a closed interval by a class of functions with a uniformly bounded third derivative is considered. It is shown that the value of best approximation of a function by this class cannot be estimated linearly in terms of its third-order modulus of continuity. At the same time, such estimates exist for classes with bounded first or second derivatives.     

A note on uniform exponential convergence of sample average approximation of random functions

Shapiro and Xu (2008) [17] investigated uniform large deviation of a class of Hölder continuous random functions. It is shown under some standard moment conditions that with probability approaching one at exponential rate with the increase of sample size, the sample average approximation of the random function converges to its expected value uniformly over a compact set. This note extends the result to a class of discontinuous functions whose expected values are continuous and the Hölder continuity may be violated for some negligible random realizations. The extension entails the application of the exponential convergence result to a substantially larger class of practically interesting functions in stochastic optimization.     

A STUDY ON GRADIENT BLOW UP FOR VISCOSITY SOLUTIONS OF FULLY NONLINEAR, UNIFORMLY ELLIPTIC EQUATIONS

We investigate sharp conditions for boundary and interior gradient estimates of continuous viscosity solutions to fully nonlinear,uniformly elliptic equations under Dirichlet boundary conditions. When ...     

Null controllability of viscous Hamilton–Jacobi equations

We study the problem of null controllability for viscous Hamilton–Jacobi equations in bounded domains of the Euclidean space in any space dimension and with controls localized in an arbitrary open nonempty subset of the domain where the equation holds. We prove the null controllability of the system in the sense that, every bounded (and in some cases uniformly continuous) initial datum can be driven to the null state in a sufficiently large time. The proof combines decay properties of the solutions of the uncontrolled system and local null controllability results for small data obtained by means of Carleman inequalities. We also show that there exists a waiting time so that the time of control needs to be large enough, as a function of the norm of the initial data, for the controllability property to hold. We give sharp asymptotic lower and upper bounds on this waiting time both as the size of the data tends to zero and infinity. These results also establish a limit on the growth of nonlinearities that can be controlled uniformly on a time independent of the initial data.     

Optimal embedding and sharp estimates of the continuity envelope for generalized Bessel potentials

In the theory of function spaces it is an important problem to describe the differential properties for the convolution u = G * f in terms of the behavior of kernel near the origin, and at the infinity. In our paper the differential properties of convolution are characterized by their modulus of continuity of order k ∈ N in the uniform norm. The kernels of convolution generalize the classical kernels determining the Bessel and Riesz potential. They admit non-power behavior near the origin. The order-sharp estimates are obtained for moduli of continuity of the convolution in the uniform norm as well as for continuity envelope function of generalized Bessel potentials. Such estimates admit sharp embedding theorems into a Calderon space and imply estimates for the approximation numbers of the embedding operator.     

Cosine families and semigroups really differ

We reveal three surprising properties of cosine families, distinguishing them from semigroups of operators: (1) A single trajectory of a cosine family is either strongly continuous or not measurable. (2) Pointwise convergence of a sequence of equibounded cosine families implies that the convergence is almost uniform for time in the entire real line; in particular, cosine families cannot be perturbed in a singular way. (3) A non-constant trajectory of a bounded cosine family does not have a limit at infinity; in particular, the rich theory of asymptotic behaviour of semigroups does not have a counterpart for cosine families. In addition, we show that equibounded cosine families that converge strongly and almost uniformly in time may fail to converge uniformly.     

STRONGLY SEQUENTIALLY CONTINUOUS FUNCTIONS

Abstract

Given a topological abelian group G, we study the class of strongly sequentially continuous functions on G. Strong sequential continuity is a property intermediate between sequential continuity and uniform sequential continuity, which appeared naturally in the study of smooth functions on Banach spaces. In this paper, we shall mainly concentrate on the gap between strong sequential continuity and uniform sequential continuity. It turns out that if G has some completeness property—for example, if it is completely metrizable—then all strongly sequentially continuous functions on G are uniformly sequentially continuous. On the other hand, we exhibit a large and natural class of groups for which the two notions differ. This class is defined by a property reminiscent of the classical Dirichlet theorem; it includes all dense sugroups of R generated by an increasing sequence of Dirichlet sets, and groups of the form (X, w), where X is a separable Banach space failing the Schur property. Finally, we show that the family of bounded, real-valued strongly sequentially continuous functions on G is a closed subalgebra of l∞(G).     

Unique solutions

It is folklore that if a continuous function on a complete metric space has approximate roots and in a uniform manner at most one root, then it actually has a root, which of course is uniquely determined. Also in Bishop's constructive mathematics with countable choice, the general setting of the present note, there is a simple method to validate this heuristic principle. The unique solution even becomes a continuous function in the parameters by a mild modification of the uniqueness hypothesis. Moreover, Brouwer's fan theorem for decidable bars turns out to be equivalent to the statement that, for uniformly continuous functions on a compact metric space, the crucial uniform “at most one” condition follows from its non‐uniform counterpart. This classification in the spirit of the constructive reverse mathematics, as propagated by Ishihara and others, sharpens an earlier result obtained jointly with Berger and Bridges. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Formally continuous functions on Baire space

A function from Baire space to the natural numbers is called formally continuous if it is induced by a morphism between the corresponding formal spaces. We compare formal continuity to two other notions of continuity on Baire space working in Bishop constructive mathematics: one is a function induced by a Brouwer‐operation (i.e., inductively defined neighbourhood function); the other is a function uniformly continuous near every compact image. We show that formal continuity is equivalent to the former while it is strictly stronger than the latter. The equivalence of formally continuous functions and those induced by Brouwer‐operations requires Countable Choice.     

A note on uniform strong convergence of bivariate density estimates

In this paper we consider a class of estimates of a bivariate density function f based on an independent sample of size n. Under the assumption that f is uniformly continuous, the uniform strong consistency of such estimates was first proved by Nadaraya (1970) for a large class of kernel functions. In this note we show that the assumption of the uniform continuity of f is necessary for this type of convergence.     

A property of compactness of families of functions harmonic with respect to a Markov process

Under quite general assumptions it is shown that any uniformly bounded sequence of functions, harmonic with respect to a Markov process, contains a sub-sequencewhich converges throughout phase space to some function harmonic with respect to this process. In the case of continuous processes, the demand of uniform boundedness can be replaced by the requirement of local uniform boundedness.Translated from Matematicheskie Zametki, Vol. 7, No. 1, pp. 109–115, January, 1970.     

Almost uniform convergence

The almost uniform convergence is between uniform and quasi-uniform one. We give some necessary and sufficient conditions under which the almost uniform convergence coincides on compact sets with uniform, quasi-uniform or uniform convergence, respectively. In the second section continuity of almost uniform limits is considered. Finally we characterize the set of all points at which a net of functions is almost uniformly convergent to a given function.     

半平面中解析函数的积分表示及在逼近中的应用

在该文中, 作者证明了满足一定增长性条件的右半平面上的解析函数可以由它在边界上的积分和其加权Blaschke乘积的和表示, 作为应用, 作者还考虑了指数多项式在实数轴上加权 Banach 空间C_α 中的完备性.     

一类二元函数连续性的等价刻画及在三角模上的应用

关于二元函数的连续,经典数学分析中有熟知的结果,即"如果二元函数连续,则必关于每个单变量连续。反之,则未必"。本文证明对于单调且对称的二元函数而言,其二元连续等价于单变量连续,并重新定义了三角模的连续。     

About the Uniform Hölder Continuity of Generalized Riemann Function

In this paper, we study the uniform Hölder continuity of the generalized Riemann function \({R_{\alpha,\beta} \,\,{\rm (with}\,\, \alpha > 1 \,\,{\rm and}\,\, \beta > 0}\)) defined by

$$R_{\alpha,\beta}(x) = \sum_{n=1}^{+\infty} \frac{\sin(\pi n^\beta x)}{n^\alpha},\quad x \in \mathbb{R},$$

using its continuous wavelet transform. In particular, we show that the exponent we find is optimal. We also analyse the behaviour of \({R_{\alpha,\beta} \,\,{\rm as}\,\, \beta}\) tends to infinity.

     

In some symmetric spaces monotonicity properties can be reduced to the cone of rearrangements

Geometric properties being the rearrangement counterparts of strict monotonicity, lower local uniform monotonicity and upper local uniform monotonicity in some symmetric spaces are considered. The relationships between strict monotonicity, upper local uniform monotonicity restricted to rearrangements and classical monotonicity properties (sometimes under some additional assumptions) are showed. It is proved that order continuity and lower uniform monotonicity properties for rearrangements of symmetric spaces together are equivalent to the classical lower local uniform monotonicity for any symmetric space over a \({\sigma}\)-finite complete and non-atomic measure space. It is also showed that in the case of order continuous symmetric spaces over a \({\sigma}\)-finite and complete measure space, upper local uniform monotonicity and its rearrangement counterpart shortly called ULUM* coincide. As an application of this result, in the case of a non-atomic complete finite measure a new proof of the theorem which is already known in the literature, giving the characterization of upper local uniform monotonicity of Orlicz–Lorentz spaces, is presented. Finally, it is proved that every rotund and reflexive space X such that both X and X* have the Kadec-Klee property is locally uniformly rotund. Some other results are also given in the first part of Sect. 2.     

指数多项式在半带形中的闭包

该文对一类复指数多项式组成的线性空间$M(\Lambda )$ 在Banach 空间 H_α中的不完备性给出了充分必要条件, 其中 H_α为在半带形 I_α={z=x+iy: x≥ 0, |y|≤α} (α > 0 )中连续, 在I_α的内部解析且当 x→∞时, f(x+ iy)在I_α中关于 y 一致地 趋向 0 的函数 f(x+ iy)全体, 其范数为上确界范数. 同时指出, 如果$M(\Lambda )$ 在 H_α中不完备, 则它的闭包cl$(M(\Lambda ))$中所有的函数都可以延拓为由Dirichlet级数表示的解析函数.