Surjectivity of Linear Differential Operators in a Weighted Space of Infinitely Differentiable Functions

We present a sufficient condition for the surjectivity of a linear differential operator of infinite order with constant coefficients in a weighted space of infinitely differentiable functions on the real line.     

On the Representation of Infinitely Differentiable Functions by Series of Exponentials

We consider the problem of representing elements of a weighted space of infinitely differentiable functions on the real line by series of exponentials.     

一个无限可微函数类的最优Lagrange 插值

本文研究\,$[-1,1]$上的一个无限可微函数类$F_\infty$在空间$L_\infty[-1,1]$及加权空间$L_{p,\omega}[-1,1]$, $1\le p< \infty$ ($\omega$是$(-1,1)$上的非负连续可积函数)的最优Lagrange插值.我们证明了基于首项系数为1且于$L_{p,\omega}[-1,1]$上有最小范数的多项式零点的Lagrange插值对$1\le p< \infty$是最优的. 同时我们给出了当结点组包含端点时的最优结点组.     

几种辅助函数方法及应用

本通过举例说明的方法说明了寻求辅助函数的几种方法及其在解题中的作用。     

On the Continuity of the Generalized Nemytskii Operator on Spaces of Differentiable Functions

We obtain sufficient conditions for the continuity of the general nonlinear superposition operator (generalized Nemytskii operator) acting from the space of differentiable functions on a bounded domain to the Lebesgue space . The values of operators on a function are locally determined by the values of both the function itself and all of its partial derivatives up to order inclusive. In certain particular cases, the sufficient conditions obtained are proved to be necessary as well. The results are illustrated by several examples, and an application to the theory of Sobolev spaces is also given.     

Calculating the derivative of piecewise functions

Exercises involving the calculation of the derivative of piecewise defined functions are common in calculus, with the aim of consolidating beginners’ knowledge of applying the definition of the derivative. In such exercises, the piecewise function is commonly made up of two smooth pieces joined together at one point. A strategy which avoids using the definition of the derivative is to find the derivative function of each smooth piece and check whether these functions agree at the chosen point. Showing that this strategy works together with investigating discontinuities of the derivative is usually beyond a calculus course. However, we shall show that elementary arguments can be used to clarify the calculation and behaviour of the derivative for piecewise functions.     

一类不可微优化的Fritz-John条件

基于星形集空间的性质,定义一类星形可微函数.这类函数是方向可微的,其方向导数可以表示成两个正齐次非负连续函数之差,其星形微分为一星形集对.对于含有不等式约束条件的星形可微优化问题,给出一个Fritz-John形式的最优性必要条件.     

一类二元函数连续与可微条件的归纳与推广

多元函数的可导性、连续性、可微性是多元函数微分学的重要内容.本文基于多元微分学教学实践,归纳了一类二元函数的可偏导、连续、可微的充分条件,并对此类二元函数做一般性推广,得到了推广函数连续与可微的充分条件.     

“复数域上微分中值定理新证”注记

所谓“实数域上的微分中值定理可以推广到复数域上”的论断值得商榷．通过给出实例的方法,具体分析所谓的“解析函数的微分中值定理”之错误所在．     

随机模糊集及其落影大数定理

本文给出随机模糊集的定义,并证明了随机模糊集的落影大数定理,为模糊集值统计提供了理论依据。     

算子函数的Weyl定理及其稳定性

设H是无限维复Hilbert空间,B(H)表示H上的有界线性算子全体构成的集合.本文对B(H)中使得f(T)满足Weyl定理的算子进行刻画,其中f是T的谱集的某个邻域上的解析函数.同时,也对算子函数的Weyl定理及算子Weyl定理的摄动之间的关系进行了讨论.     

多复变量亚纯函数的第二基本定理和唯一性定理

本文给出了多复变量亚纯函数的第二基本定理的一个改进形式,并进一步得到了一个关于重值的唯一性定理(详见文[1-16])。     

线性空间上凸函数的可微性与次可微性

讨论了定义在线性空间上的广义实值凸函数的可做性和次可微性．     

Function S-rough Sets and Their Law Characteristic

Function S-rough sets(Function Singular rough sets) are defined by R-function equivalence class which has dynamic characteristic, and a function is s law, function S-rough sets have law characteristic. Function S-rough sets has these forms: function one direction S-rough sets, function two direction S-rough sets and dual of function one direction S-rough sets. This paper presents the law characteristic of function one direction S-rough sets and puts forward the theorems of law-chain-attribute and law-belt. Function S-rough sets is s new research direction of the rough sets theory.     

Generalized Besov spaces and Triebel-Lizorkin spaces

In this paper the classical Besov spaces Bsp.q and Triebel-Lizorkin spaces Fsp.q for s ∈R are generalized in an isotropy way with the smoothness weights {|2j|aln}∞j=0. These generalized Besov spaces and Triebel-Lizorkin spaces, denoted by Bap.q and Fap.q for a ∈Irk and k ∈N, respectively, keep many interesting properties, such as embedding theorems (with scales property for all smoothness weights), lifting properties for all parameters a, and duality for index 0 < p < ∞. By constructing an example, it is shown that there are infinitely many generalized Besov spaces and generalized Triebel-Lizorkin spaces lying between Bs,p.q and ∪tsBt,p.q,and between Fsp.q and ∪ts Ftp.q, respectively. Between Bs,p,q and ∪tsBt,p.qq,and between Fsp,qand ∪tsFtp.q,respectively.     

具有公共值集和亏值的亚纯函数唯一性

§ 1 .　IntroductionandtheMainResult　　Inthispaper,weshallusethestandardnotationsofvaluesdistributiontheory(see [1]) .LetfbeameromorphicfunctionandSbeasetofdistinctelements.SetEf(S) =∪a∈S{z:f =a ,countingmultiplicities} .Recently,manygoodresultswereobtained(see [1— 4]) ,whichansweredthequestionofGross(see [5 ]) .TheoremA(see [3])　LetS1 ={ 0 } ,S2 ={z :z2 (z+a) -b =0 } ,wherea,baretwononzeroconstantssuchthatthealgebraicequationz2 (z+a) -b=0hasnomultipleroots.Iff ,garetwoentirefunctio…     

Isotropic Hypersurfaces and Minimal Extensions of Lipschitz Functions

The existence and uniqueness theorem for isotropic hypersurfaces with prescribed boundary in Lorentzian warped products is proved.The proof is based on minimal Lipschitz extensions of functions. __________ Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 39, No. 3, pp. 28–36, 2005 Original Russian Text Copyright #x00A9; by A. A. Klyachin and V. M. Miklyukov     

Growth of Entire Functions with Given Zeros and Representation of Meromorphic Functions

Mathematical Notes - Let $\Lambda = \{ \lambda n\}$ be a sequence of points on the complex plane, and let $\Lambda (r)$ be the number of points of the sequence $\Lambda$ in the disk $\{...     

Characteristic Functions and Symbols in the Theory of Feller Processes

We derive a probabilistic expression for the symbol of the generator of a Feller process.     

Holomorphic Functions Having Large Images under the Action of Differential Operators

We prove in this note that, given a simply connected domainGin the complex plane and a sequence of infinite order linear differential operators generated by entire functions of subexponential type satisfying suitable conditions, then there are holomorphic functionsfonGsuch that the image of any open subset under the action of those operators onfis arbitrarily large. This generalizes an earlier result about images of derivatives. A known statement about close orbits is also strengthened.