本文選題：Fourier級數(shù)　切入點：均值有界變差　出處：《浙江理工大學》2016年碩士論文　論文類型：學位論文

【摘要】：在分析學的研究領域中,三角級數(shù)有著非常重要的作用并且在其他相關的科學和工程領域也有許多重要的應用.因此,在很早以前許多學者就開始關注三角級數(shù)的收斂性并對其進行研究.研究三角級數(shù)的收斂性,首先要考慮它的系數(shù)問題.系數(shù)的單調性條件的推廣有長久的歷史,單調性不斷被推廣到各種有界變差條件,最終,推廣到均值有界變差(MVBV)條件.隨后,人們對三角積分的研究也產(chǎn)生了很大興趣.本文在前人研究三角級數(shù)的基礎上,將系數(shù)數(shù)列的MVBV條件推廣到函數(shù)的MVBV條件,并研究正弦和余弦積分在MVBV條件下的加權可積性問題.文中共分為四章：第一章緒論本章追溯了三角級數(shù)可積性問題的歷史,簡要介紹了其發(fā)展現(xiàn)狀,并給出論文中常用的符號和定義.第二章MVBV函數(shù)類的加權可積性Wang和Zhou在2010年對Boas-Heywood定理在MVBV條件下做了相應的推廣.基于此條件,本章將結論推廣到MVBV函數(shù)類,對非負的正弦和余弦積分給出了充分必要條件.第三章實意義下的MVBV函數(shù)類的加權可積性在取消非負性的基礎上本章繼續(xù)對MVBV函數(shù)的加權可積性進行研究.采用了不同于前一章定理證明的方法和技巧.我們證明了：假設0α1, f(x)∈MVBVF是[0,∞)上的有界變差實函數(shù),對于任意的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)的正弦積分.一個相應的逆定理也在本章得以建立.第四章總結本章對全文進行總結和展望.
[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 偽 1, f X) 鈭，

本文編號：1643129