若干经典定理在实有界变差条件下的推广

发布时间：2018-03-06 03:11

本文选题：级数　切入点：数列　出处：《浙江理工大学》2016年硕士论文　论文类型：学位论文

【摘要】：在分析学中,三角级数和Fourier级数系数的单调递减条件及其推广是人们研究的焦点之一.1916年,英国学者Chaundy和Jolliffe对三角级数一致收敛性在单调条件下建立了一个经典定理.随后,许多学者继续了这个方面的工作.单调性由此推广到各种拟单调和各种有界变差条件之下.2005年,乐瑞君和周颂平提出了兼容这两个方向的分组有界变差(GBV)的概念.后来,在最一般的均值有界变差概念提出之后,周颂平等人于2014年又将此推广到了实意义条件下并在经典分析中建立了许多重要的应用.本文在前人的基础上,对柯西并项准则、强逼近及其相关嵌入定理进行进一步推广研究.在研究过程中,将原有经典定理中的条件推广至实意义条件下,对复杂变号的区间采用巧妙的分割方法,以此来证明两个定理并说明定理的最终适用范围.此外,顺便给出了几乎单调递减数列与分组有界变差数列互不包含的关系.全文共分为五章：第一章为绪论,首先介绍国内外研究现状,接着对论文涉及的相关符号一一给出定义,并阐述各种数列的概念.最后,简单地介绍论文的结构框架.第二章针对Otto Szasz的柯西收敛准则进行推广.此前,乐瑞君和解烈军已经将该定理推广至非负的分组有界变差(GBV)条件并证明分组有界变差的不可减弱性.本章将经典柯西并项准则推广的条件减弱到实意义下的分组有界变差(GBV*)条件,采用特殊的分割方法建立数列及积分的相关定理.同时,举例应用该定理.在第三章中,主要对Tikhonov在拟单调(QM)及剩余有界变差(RBV)条件下的强逼近及其相关嵌入定理进行研究.2010年,王敏芝已经给出单边、非负的均值有界变差(MVBV)条件下的定理.本章在第二章分割的基础上进一步细分,对分割作出了本质性的推广.最终,我们给出了实意义下修正的均值有界变差的强逼近及其相关嵌入定理的证明.虽然几乎单调递减数列(AMS)与分组有界变差数列(GBVS)之间互不包含的关系是显而易见的,但这需要一个具体的证明过程.第四章将通过构造平凡与非平凡数列的反例来证明此关系.最后,我们对全文进行总结与展望.
[Abstract]:In the field of analysis, the monotone decreasing condition and its generalization of the coefficients of trigonometric series and Fourier series are one of the focuses of research. In 1916, the British scholars Chaundy and Jolliffe established a classical theorem on the uniform convergence of trigonometric series under monotonic conditions. Many scholars have continued their work in this field. Monotonicity is thus extended to various quasi-monotone and bounded variation conditions. In 2005, Le Ruijun and Zhou Songping put forward the concept of bounded variation in groups compatible with these two directions. After the most general concept of bounded mean variation was put forward, Zhou Songping and others extended it to the real meaning condition in 2014 and established many important applications in classical analysis. The strong approximation and its related embedding theorems are further generalized and studied. In the course of the research, the conditions in the original classical theorems are extended to the real meaning conditions, and the subdivision method is used for the interval of complex sign variation. In addition, the relation between the sequence of almost monotone decreasing numbers and the sequence of bounded variable number of groups is given. The whole paper is divided into five chapters: the first chapter is the introduction. This paper first introduces the current research situation at home and abroad, then defines the relevant symbols involved in the paper, and expounds the concepts of various series of numbers. In chapter 2, the Cauchy convergence criterion of Otto Szasz is generalized. The theorem has been extended to the condition of nonnegative bounded variation of grouping (GBV) and proved the irabligibility of bounded variation of grouping. In this chapter, the condition of the extension of the classical Cauchy complex criterion is reduced to the grouping in the real sense. Bounded variation condition, A special partition method is used to establish the relevant theorems of sequence and integral. At the same time, an example is given to apply the theorem. In this paper, the strong approximation and its related embedding theorems of Tikhonov under the condition of quasi monotone QM) and residual bounded variation are studied. In 2010, Wang Minzhi presented one-sided approximation. Theorem under the condition of bounded variation of nonnegative mean value MVBV). In this chapter, the segmentation is further subdivided on the basis of the second chapter, and the essential generalization of the segmentation is made. In this paper, we give the proof of the strong approximation of the modified mean bounded variation and its related embedding theorem in the real sense. Although the relation between the almost monotone decreasing sequence (AMS) and the grouped bounded variable difference sequence (GBVS) is obvious, However, this requires a concrete proof process. Chapter 4th will prove this relationship by constructing counterexample of ordinary and nontrivial sequence. Finally, we summarize and look forward to the full text.
【学位授予单位】：浙江理工大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O173

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