本文選題：傅里葉級(jí)數(shù)　切入點(diǎn)：GBVS　出處：《浙江理工大學(xué)》2015年碩士論文　論文類型：學(xué)位論文

【摘要】：三角級(jí)數(shù)論是一個(gè)龐大的數(shù)學(xué)領(lǐng)域,他包含F(xiàn)ourier分析中位于基礎(chǔ)地位的Fourier級(jí)數(shù).其中Chaundy和Jolliffe在單調(diào)性和非負(fù)性條件下證明了正弦級(jí)數(shù)的一致收斂性.之后研究者們將單調(diào)性條件逐步推廣到一些擬單調(diào)條件上,如：擬單調(diào)條件,正則變化擬單調(diào)條件和O-正則變化擬單調(diào)條件. 匈牙利數(shù)學(xué)家Leindler在2001年將注意力轉(zhuǎn)移到剩余有界變差的概念上來推廣單調(diào)性條件.然而,在2002年他證明了剩余有界變差條件和O-正則變化擬單調(diào)條件是互不包含的.之后,樂瑞君和周頌平在2005年定義了包含剩余有界變差概念和O-正則變化擬單調(diào)概念的分組有界變差概念,最終,周頌平等在2010年給出了均值有界變差的概念.大量經(jīng)典結(jié)果,如正余弦級(jí)數(shù)的一致收斂性,Fourier級(jí)數(shù)的L1-收斂性和Lp可積性等均被推廣到了均值有界變差條件上. 在Zygmund的書"Trigonometric Series"中證明了正余弦級(jí)數(shù)的漸近公式,并由Hardy將其推廣到單調(diào)性條件下并給出了漸近公式的充分必要條件,之后人們建立了一些相應(yīng)的推廣.在1992年,Nurcomb將漸近公式推廣到擬單調(diào)條件上.有趣的是,謝庭藩和周頌平在1994年證明了漸近公式的充分性部分在O-正則變化擬單調(diào)條件下不再成立,而必要性部分則需要加強(qiáng).后來,樂瑞君,周頌平,王敏之和趙易將漸近公式推廣到分組有界變差和均值有界變差條件,同時(shí)證明了L2π-可積性. 由Leindler的文章[8]獲得啟發(fā),我們在論文開始研究了這些概念之間的關(guān)系.我們知道Fourier變換在計(jì)算和工程學(xué)上有著重要的應(yīng)用,本論文的第二個(gè)目標(biāo)是建立Fourier變換中的相應(yīng)結(jié)果. 全文共分為四章來闡述： 第一章中主要給出這些問題已有的相關(guān)背景和工作,并列舉了一些相關(guān)的定義和包含關(guān)系. 在第二章中,從樂瑞君和周頌平的定理和Leindler的工作開始,我們證明了均值有界變差條件與分組有界變差條件在條件下是等價(jià)的,進(jìn)一步我們構(gòu)造反例證明了條件(1.2)不能省去,否則漸近等式不能保持成立.另外,我們也研究了分組有界變差數(shù)列,O-正則變化擬單調(diào)數(shù)列和擬單調(diào)數(shù)列之間的等價(jià)關(guān)系. 在第三章中,我們考慮了均值有界變差函數(shù)并給出了一組Fourier變換的漸近公式,同時(shí)證明了均值有界變差函數(shù)與O-正則變化擬單調(diào)函數(shù)之間的等價(jià)關(guān)系. 在第四章中,我們推廣了第二章中的一個(gè)有用的引理,并給出一個(gè)精細(xì)的反例證明均值有界變差條件在這些漸近公式中不能取消.
[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.
【學(xué)位授予單位】：浙江理工大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類號(hào)】：O173

【共引文獻(xiàn)】

相關(guān)期刊論文 前3條

1 ZHOU SongPing;FENG FenJun;ZHANG LiJun;;Trigonometric series with piecewise mean value bounded variation coefficients[J];Science China(Mathematics);2013年08期

2 周頌平;樂瑞君;;單調(diào)性條件在Fourier級(jí)數(shù)收斂性中的最終推廣:歷史、發(fā)展、應(yīng)用和猜想[J];數(shù)學(xué)進(jìn)展;2011年02期

3 張麗君;;三角級(jí)數(shù)一致收斂性問題在復(fù)空間的完整推廣[J];數(shù)學(xué)雜志;2012年03期

相關(guān)碩士學(xué)位論文 前4條

1 張麗君;均值有界變差條件的進(jìn)一步推廣[D];浙江理工大學(xué);2012年

2 馮奮軍;Fourier級(jí)數(shù)的L^1收斂性[D];浙江理工大學(xué);2012年

3 夏星星;分析中經(jīng)典不等式和Fourier積分一致收斂性的推廣研究[D];浙江理工大學(xué);2014年

4 張靜;基于分段條件下的正弦積分和余弦積分性質(zhì)研究[D];浙江理工大學(xué);2014年

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本文編號(hào)：1602351