The Related Results Of Sine And Cosine Integrals Under The Piecewise Condition

As the core content of Fourier analysis, trigonometric series have attracted a lot of in-terests of mathematicians. The study of the convergence issue of Fourier analysis originated from the classic Chaundy-Jollife’s theorem. With the continuous efforts of many mathe-maticians, monotonicity setting on the coefficients has been weakened finally to the MVBV condition. In order to cancel nonnegativity, Zhou proposed PBV condition and obtained the uniform convergence of trigonometric series, integrability and relevant results. Later, Zhou further proposed PMBV condition, meanwhile F Moricz proposed trigonometric transform type definition of MVBV condition and established the uniform convergence theorem of sine integrals. Based on these two methods, this paper respectively presents the definitions of functions under PBV condition and PMBV condition, and establishes uniform convergence and uniform boundedness theorems of sine and cosine integrals. This thesis is divided into four chapters:Chapter One:IntroductionThis chapter gives a brief survey on the related background of the coefficient conditions of trigonometric series, convergence and some related results of trigonometric transform.Chapter two:Convergence of sine and cosine integrals under the PBV conditionThis chapter’s content is to study the uniform convergence of sine and cosine Fourier transforms under the condition of PBV. By giving the definition of PBV condition of func-tions and establishing related lemmas, we will prove two important theorems and a corollary, finally give the property of uniform boundedness.Chapter three:Convergence of sine and cosine integrals under the PMBV conditionThis chapter continues to study the uniform convergence of sine and cosine Fourier transforms under the condition of PMBV. Corresponding results are established to the pre-vious chapter.Chapter four:SummaryThis chapter gives a summary and an outlook in the end.