Research And Extension Of The Classical Inequalities And The Uniform Convergence Of Fourier Integral In Analysis

With the rise of analysis, trigonometric series as a powerful mathematical tool are greatly concerned. From1807to1822, Fourier series are produced because of the applications in Physics, especially widely used in thermal conductivities and chord vibration. However, the people find it is almost impossible to calculate the series accurately, so that they have to consider the convergence problem in order to do the approximation. Among the problems of the uniform convergence and the mean convergence, the generalization of monotonicity condition setting on the coefficients of trigonometric series and Fourier series are the focus concerned.Under nonnegativity and monotonicity condition, in1916, Chaundy and Jolliffe first established a classical result of uniform convergence of trigonometric series. Later, various generalizations such as quasi-monotone (QM) condition, rest bounded variation (RBV) con-dition, group bounded variation (GBV)condition, non-onesided bounded variation (NBV) condition and, ultimately, mean value bounded variation (MVBV) condition are produced in go years.This paper will study related fields on the basis of previous works. First, the trigono-metric inequality which is often used in Fourier analysis is generalized to expand the scope of applications. In addition, we give a mending to a little but sensitive flaw in the original proof of an important and useful inequality established by Leinder. Finally, based on Zhang and Ko-rus’conclusions, the uniform convergence of trigonometric series in complex space has been extended to the integral form. In other words, this paper establishes the uniform convergence theorem of Fourier integrals in the complex space.The full thesis is divided into four chapters:The first chapter is an introduction, the background of research contents and the current development are figured out here. Then symbols, definitions on the paper involved are also defined. In addition, generalized various monotonicity sequences and the relationships between them are listed.In the second chapter, we weaken the monotone condition of the classic gonometric inequality which was first established by Telyakovskii. Although Wang and Zhao extended it to the mean value bounded variation (MVBV) condition, we weaken the condition of the classic trigonometric inequality to the second supremum bounded variation (SBV2) condition and construct a series which is not in the MVBVS but in SBVS2.The flaw of a inequality generalized by Leindler has been indicated in the third chaper. To mend this falw, we establish a supplementary theorem.In the fourth chapter, we generalize the convergence theorem of trigonometric series to the integral form and establish the convergence of Fourier integral in the complex space.