无穷限反常积分收敛时被积函数的极限状态

根据无穷限反常积分∫a^＋∞f（x）dx收敛的柯西准则和定积分的性质,讨论被积函数f（x）当x→∞时。的极限状态,并得出当无穷限反常积分∫a^＋∞f（x）dx收敛且f（x）在[a,＋∞）上连续,或者无穷限反常积分∫a^＋∞f（x）dx绝对收敛时,存在数列{xn}∩[a,＋∞]且xn→＋∞（n→∞）,使limn→∞xnf（xn）=0.     

反常积分敛散性的根值判别法

由于积分与级数在理论上是统一的,因此有关正项级数的根式判别法可被推广以判别无穷限积分和瑕积分的敛散性.设f(x)是[a,+∞)上的非负函数,li mx→+∞xf(x)=ρ,则当ρ1时,反常积分∫a+∞f(x)dx收敛,而当ρ1时,反常积分∫a+∞f(x)dx发散;设f(x)是(a,b]上的非负函数,a为瑕点,xli→ma+(f(x))x-a=ρ,则当ρ1时,反常积分∫abf(x)dx收敛,而当ρ1时,反常积分∫baf(x)dx发散.     

极限lim x→＋∞f（x）=0的一个充分条件的探讨

从无穷积分∫＋∞ a f（x）dx收敛与无穷远极限lim x→＋∞f（x）=0之间的关系展开论述,研究在广义积分∫＋∞ a f（x）dx收敛的前提下,无穷远极限lim x→＋∞f（x）=0的一个充分条件．在此基础上,适当减弱条件得到该条件的推广形式,为更好的解决无穷远极限lim x→＋∞f（x）=0的问题提供更一般的方法．     

正函数广义积分收敛性的一个判别法

研究正函数广义积分的敛散性．利用二重积分的性质．从被积函数自身的性态出发．当自变量x充分大时,通过讨论∫β（x＋σ）^β（x＋σ＋1）f（y）dy与f（x）的比值（其中β≥1,σ∈R为固定常数）,可建立一个收敛判别法．并可平行给出相应正项级数审敛法。此法是对DAlembert审敛法和双比值审敛法的推广．     

反常积分∫^∞x sint/t^αdt的估计式

当O〈a〈2时,积分∫^∞x sint/t^αdt收敛．本文研究在2≤a〈4时,反常积分∫^∞x sint/t^αdt当x→0^＋时的估计式．     

关于无穷积分收敛的必要条件的讨论

当无穷积分∫0^ ∞f(x)dx收敛时,若f（x)在[0, ∞]上一致连续,或者知lim x→ ∞f(x)存在,那么都有lim x→ ∞f(x)=0。     

关于反常积分的一个注记

本文讨论无穷限反常积分∫_a~(+∞)f(x)dx收敛的必要条件.首先考虑黎曼积分意义下该反常积分收敛的必要条件,其结果包含了刘妮、刘卫江的工作(见《高等数学研究》,17卷,6期,6-9页,2014年11月).然后研究勒贝格积分意义下∫_a~(+∞)f(x)dx存在的必要条件.     

∫a＋∞f（x）dx收敛与limx→＋∞f（x）=0

讨论了当广义积分∫a ∞f(x)dx收敛时,极限linx→ ∞f(x)=0的各种条件。     

对一类反常积分收敛判别题的研究

命题1在p>0的前提下,讨论反常积分+∞∫0esinxsin2xpxdx的敛散性.从题解角度讲,命题1着重讨论0与正无穷两处奇点的敛散性,为此将原积分改写为2π∫0esinxsin2xpxdx++∞∫2πesinxsin2xpxdx.由于能找到esin∫xsin2xdx在R上有界的原函数2(sinx-1)esinx,故可以帮助着手x趋于正无穷时的敛散性讨论.笔者欲对命题1作推广,即寻找一类正实数a,使命题1的解法在“a”代替sin2x中的“2”时可以沿用.先于0处讨论反常积分2π∫0esinxsinaxpxdx的散性.命题2对任何给定正整数a,反常积分2π∫0esinaxsin2xpxdx当p 2时发散,当0相似文献   

积分第二中值定理中间点函数的连续性及可微性

主要研究按积分第二中值定理∫a^xf（t）g（t）dt=f（a）∫a^ξg（t）dt＋f（x）∫ξxg（t）dt确定的中间点ξ作为x的函数,其连续性及可微性．     

RECENT PROGRESS ON SPHERICAL HARMONIC APPROXIMATION MADE BY BNU RESEARCH GROUP ——In memory of Professor Sun Yongsheng

As early as in 1990, Professor Sun Yongsheng, suggested his students at Beijing Normal University to consider research problems on the unit sphere. Under his guidance and encouragement his students started the research on spherical harmonic analysis and approximation. In this paper, we incompletely introduce the main achievements in this area obtained by our group and relative researchers during recent 5 years （2001-2005）. The main topics are： convergence of Cesaro summability, a.e. and strong summability of Fourier-Laplace series; smoothness and K-functionals; Kolmogorov and linear widths.     

Instructions for Authors

<正>Submission Authors must use LaTeX for typewriting,and visit our website www.actamath.com to submit your paper.Our address is Editorial Office of Acta Mathematica Sinica,Academy of Mathematics and Systems Science,Chinese Academy of Sciences,Beijing 100190,P.R.China.     

THE 8~(th) INTERNATIONAL CONGRESS ON INDUSTRIAL AND APPLIED MATHEMATICS

正August 10-14,2015Beijin,China The International Congress on Industrial and Applied Mathematics(ICIAM)is the premier international congress in the field of applied mathematics held every four years under the auspices of the International Council for Industrial and Applied Mathematics.From August 10 to 14,2015,mathematicians,scientists     

Boundedness for commutators of multipliers on Hardy type spaces

In this paper, we study the commutators generalized by multipliers and a BMO function. Under some assumptions, we establish its boundedness properties from certain atomic Hardy space Hb^p（R^n） into the Lebesgue space L^p with p 〈 1.     

A unified approach to certain problems of best local approximation

In this paper we study best local quasi-rational approximation and best local approximation from finite dimensional subspaces of vectorial functions of several variables. Our approach extends and unifies several problems concerning best local multi-point approximation in different norms.     

Towards Excellence in Science Chinese Academy of Sciences

<正>May 26,2014,Beijing Science is a human enterprise in the pursuit of knowledge.The scientific revolution that occurred in the 17th Century initiated the advances of modern science.The scientific knowledge system created by     

THE 8~(th) INTERNATIONAL CONGRESS ON INDUSTRIAL AND APPLIED MATHEMATICS

<正>August 10-14,2015Beijing,ChinaThe International Congress on Industrial and Applied Mathematics(ICIAM)is the premier international congress in the field of applied mathematics held every four years under the auspices of the International Council for Industrial and Applied Mathematics.From August 10 to 14,2015,mathematicians,scientists     

Bounds for the zeros of polynomials

Let P(z)=∑↓j=0↑n ajx^j be a polynomial of degree n. In this paper we prove a more general result which interalia improves upon the bounds of a class of polynomials. We also prove a result which includes some extensions and generalizations of Enestrǒm-Kakeya theorem.