L ^ 1 Convergence Of Fourier Series

Trigonometric series as a powerful mathematical tool have a wide range of applicationsin many fields. To deal with trigonometric series (Fourier series) efciently, it is importantto study their convergence at first. Especially, in the uniform convergence and mean conver-gence of trigonometric (Fourier) series, scholars are interested in the ultimate generalizationof monotonicity on the coefcients. In this thesis, we generalized Korus’s results under hisGM7condition setting upon the Fourier series with nonnegative coefcients to complex spacein a non-trivial way with some kind of balance conditions. For these Fourier series with gen-eral coefcients (may have diferent signs), we give a essential generalization on the classicalresult of L1-convergence under our new PMBV condition. We also study the L1-integrabilityof sine (cosine) trigonometric series with coefcients satisfying PMBV condition.This thesis is divided into five chapters.In the first chapter, we introduced the background of the research contents and thestatus of current studies. We also discuss relationship among various works. we will givesome symbols and notations frequently used.People are interested in conditions what can generalize monotonicity setting upon thecoefcients which guarantee the classical results of trigonometric series. Although the MVBVcondition is ultimate in some sense, from the point of view of set theory, questions still persisthow the “boundary points” of MVBVS can behave. The GM7condition is a meaningful tryto generalize MVBV condition in the “boundary points”. Our work in the second chapteris to generalize Korus’s results under the GM7condition to the complex space in a non-trivial way with the one-side condition and some kind of balance conditions. Especially, ourgeneralization also mends the neglect left in all precedents, i.e., the results in the complexspace should include sine and cosine series cases. It may have some significance to laterworks of this kind.For trigonometric (Fourier) series with general coefcients (may have diferent signs),the initial results were established for rarely changing coefcients. Recently, Zhou gave anessential generalization to PBV condition. In this charpter, we propose a new conditionPMBV, and prove that this is a right real generalization of PBV, as well as MVBV. We thenestablished the classical result on the L1-convergence of Fourier series whose coefcientssatisfy our new PMBV condition.In classical analysis, an interesting question is what condition imposed on coefcients of sine or cosine trigonometric series can guarantee the integrability for the sum function ofthe series. The class result was established under monotonicity setting on the coefcients.This chapter will give it a generalization under PMBV condition.Finally, we summarize and discuss some problems for future research.