对分析中一个重要渐近等式的推广

发布时间：2018-03-12 16:25

本文选题：傅里叶级数　切入点：GBVS　出处：《浙江理工大学》2015年硕士论文　论文类型：学位论文

【摘要】：三角级数论是一个庞大的数学领域,他包含Fourier分析中位于基础地位的Fourier级数.其中Chaundy和Jolliffe在单调性和非负性条件下证明了正弦级数的一致收敛性.之后研究者们将单调性条件逐步推广到一些拟单调条件上,如：拟单调条件,正则变化拟单调条件和O-正则变化拟单调条件. 匈牙利数学家Leindler在2001年将注意力转移到剩余有界变差的概念上来推广单调性条件.然而,在2002年他证明了剩余有界变差条件和O-正则变化拟单调条件是互不包含的.之后,乐瑞君和周颂平在2005年定义了包含剩余有界变差概念和O-正则变化拟单调概念的分组有界变差概念,最终,周颂平等在2010年给出了均值有界变差的概念.大量经典结果,如正余弦级数的一致收敛性,Fourier级数的L1-收敛性和Lp可积性等均被推广到了均值有界变差条件上. 在Zygmund的书"Trigonometric Series"中证明了正余弦级数的渐近公式,并由Hardy将其推广到单调性条件下并给出了渐近公式的充分必要条件,之后人们建立了一些相应的推广.在1992年,Nurcomb将渐近公式推广到拟单调条件上.有趣的是,谢庭藩和周颂平在1994年证明了渐近公式的充分性部分在O-正则变化拟单调条件下不再成立,而必要性部分则需要加强.后来,乐瑞君,周颂平,王敏之和赵易将渐近公式推广到分组有界变差和均值有界变差条件,同时证明了L2π-可积性. 由Leindler的文章[8]获得启发,我们在论文开始研究了这些概念之间的关系.我们知道Fourier变换在计算和工程学上有着重要的应用,本论文的第二个目标是建立Fourier变换中的相应结果. 全文共分为四章来阐述：第一章中主要给出这些问题已有的相关背景和工作,并列举了一些相关的定义和包含关系. 在第二章中,从乐瑞君和周颂平的定理和Leindler的工作开始,我们证明了均值有界变差条件与分组有界变差条件在条件下是等价的,进一步我们构造反例证明了条件(1.2)不能省去,否则渐近等式不能保持成立.另外,我们也研究了分组有界变差数列,O-正则变化拟单调数列和拟单调数列之间的等价关系. 在第三章中,我们考虑了均值有界变差函数并给出了一组Fourier变换的渐近公式,同时证明了均值有界变差函数与O-正则变化拟单调函数之间的等价关系. 在第四章中,我们推广了第二章中的一个有用的引理,并给出一个精细的反例证明均值有界变差条件在这些渐近公式中不能取消.
[Abstract]:The theory of trigonometric series is a huge field of mathematics. He includes Fourier series in Fourier analysis, in which Chaundy and Jolliffe prove the uniform convergence of sinusoidal series under monotonicity and non-negativity conditions. Then the monotonicity conditions are extended to some quasi-monotone conditions. For example: quasi monotone condition, regular variation quasi monotone condition and O- regular variation quasi monotone condition. In 2001, the Hungarian mathematician Leindler shifted his attention to the concept of residual bounded variation to generalize the monotonicity condition. However, in 2002, he proved that the condition of residual bounded variation and the quasi-monotone condition of O- regular variation were not included. In 2005, Le Ruijun and Zhou Songping defined the concepts of bounded variation in groups containing the concepts of residual bounded variation and quasi-monotone of O- regular variation. Finally, Zhou Songping and others gave the concept of bounded variation of mean value in 2010. For example, the uniform convergence of sine cosine series and the L 1-convergence and LP integrability of Fourier series are generalized to the mean bounded variation condition. In Zygmund's book Trigonometric Series, the asymptotic formula of sine cosine series is proved, which is extended by Hardy to monotonicity and the necessary and sufficient conditions for asymptotic formula are given. In 1992, Nurcomb extended the asymptotic formula to quasi-monotone conditions. In 1994, Xie Ting-fan and Zhou Songping proved that the sufficient part of the asymptotic formula no longer holds under the quasi-monotone condition of O- regular variation, but the necessary part needs to be strengthened. The sum of Wang Min and Zhao Yi generalized the asymptotic formula to the conditions of bounded variation of groups and bounded variation of mean value, and proved the L 2 蟺 -integrability. Inspired by Leindler's paper [8], we begin to study the relationship between these concepts. We know that Fourier transformation has important applications in computation and engineering. The second goal of this paper is to establish the corresponding results of Fourier transform. The full text is divided into four chapters to elaborate:. In the first chapter, the background and work of these problems are given, and some relevant definitions and inclusions are given. In the second chapter, starting with the theorem of Le Ruijun and Zhou Songping and the work of Leindler, we prove that the mean bounded variation condition is equivalent to the grouping bounded variation condition under the condition. Furthermore, we construct a counter example to prove that condition 1. 2) can not be omitted. Otherwise, the asymptotic equation can not hold true. In addition, we also study the equivalent relations between the group bounded variable-difference sequence and the quasi-monotone sequence of the quasi-monotone sequence. In chapter 3, we consider the mean-bounded variation function and give a set of asymptotic formulas of Fourier transformation. We also prove the equivalent relation between the mean-bounded variation function and the O-regular variation quasi-monotone function. In Chapter 4th, we generalize a useful Lemma in Chapter 2 and give a fine counterexample to prove that the bounded variation condition of mean value cannot be cancelled in these asymptotic formulas.
【学位授予单位】：浙江理工大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O173

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