在MVBV条件下的加权可积性

发布时间：2018-03-21 08:51

  本文选题：Fourier级数　切入点：均值有界变差　出处：《浙江理工大学》2016年硕士论文　论文类型：学位论文

【摘要】：在分析学的研究领域中,三角级数有着非常重要的作用并且在其他相关的科学和工程领域也有许多重要的应用.因此,在很早以前许多学者就开始关注三角级数的收敛性并对其进行研究.研究三角级数的收敛性,首先要考虑它的系数问题.系数的单调性条件的推广有长久的历史,单调性不断被推广到各种有界变差条件,最终,推广到均值有界变差(MVBV)条件.随后,人们对三角积分的研究也产生了很大兴趣.本文在前人研究三角级数的基础上,将系数数列的MVBV条件推广到函数的MVBV条件,并研究正弦和余弦积分在MVBV条件下的加权可积性问题.文中共分为四章：第一章绪论本章追溯了三角级数可积性问题的历史,简要介绍了其发展现状,并给出论文中常用的符号和定义.第二章MVBV函数类的加权可积性Wang和Zhou在2010年对Boas-Heywood定理在MVBV条件下做了相应的推广.基于此条件,本章将结论推广到MVBV函数类,对非负的正弦和余弦积分给出了充分必要条件.第三章实意义下的MVBV函数类的加权可积性在取消非负性的基础上本章继续对MVBV函数的加权可积性进行研究.采用了不同于前一章定理证明的方法和技巧.我们证明了：假设0α1, f(x)∈MVBVF是[0,∞)上的有界变差实函数,对于任意的aA1,faa+1xα|f(x)|dx一致有界.如果那么其中F(t)=∫0∞f(x)sin txdx是f(χ)的正弦积分.一个相应的逆定理也在本章得以建立.第四章总结本章对全文进行总结和展望.
[Abstract]:Trigonometric series play a very important role in the field of analytical research and have many important applications in other related fields of science and engineering. Many scholars began to study the convergence of trigonometric series long ago. In order to study the convergence of trigonometric series, first of all, the coefficient problem should be considered. The promotion of monotonicity conditions of coefficients has a long history. Monotonicity has been extended to a variety of bounded variation conditions, and finally to the mean bounded variation condition MVBV). Subsequently, people also have a great interest in the study of trigonometric integrals. In this paper, based on the previous studies of trigonometric series, The MVBV condition of coefficient sequence is extended to the MVBV condition of function, and the weighted integrability of sinusoidal and cosine integrals under MVBV condition is studied. The paper is divided into four chapters: the first chapter introduces the history of the integrability problem of trigonometric series. In chapter 2, the weighted integrability of MVBV function class Wang and Zhou generalized Boas-Heywood theorem under MVBV condition in 2010. In this chapter, the conclusion is extended to the class of MVBV functions. The necessary and sufficient conditions for nonnegative sinusoidal and cosine integrals are given. In Chapter 3, the weighted integrability of MVBV functions in the sense of reality is further studied on the basis of eliminating non-negativity. In this chapter, the weighted integrability of MVBV functions is studied. Different from the methods and techniques of theorem proof in the previous chapter, we prove that we assume that 0 伪 1, f X) 鈭，

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