A Generalization Of Some Classical Results Under Bounded Variation Conditions In Real Sense

In the analysis, the generalization of monotonicity condition setting on the coefficients of trigonometric series and Fourier series is one of the focus of the study. In 1916, Chaundy and Jolliffe first established a classical theorem of uniform convergence of trigonometric series under monotonicity condition. Later, many scholars have continued this type of work. Monotonicity is generalized to various quasi-monotonicity and various bounded variation conditions. In 2005, Le and Zhou raised the group bounded variation concept to contain both generalizations. After the ultimate mean value bounded variation, Zhou etc. generalized it to real sense in 2014, and established many important applications in classical analysis.Based on the previous works, this paper will generalize the Cauchy’s condensation criterion and the strong approximation of Fourier series and related embedding theorems. In the process, the condition of classic theorem is generalized to group bounded variation or modified mean value bounded variation in real sense respectively. They are both proved to be the ultimate conditions in general sense in each case. In addition, the relationship between almost monotone sequences and group bounded variation sequences is also illustrated.The full thesis is divided into five chapters:The first chapter is an introduction. The development history is figured out here briefly and the related symbols and definitions are also given. Finally, the structure of the thesis is outlined.The second chapter is to promote the Cauchy’s condensation criterion, originally pro-posed by Otto Szasz. Le and Xie have extended the condition to the group bounded varia-tion before. This chapter will give a remark on Cauchy’s condensation criterion under group bounded variation condition in real sense. Last but not the least, artful segmentation is adopted. Moreover, the theorem applied as an example is given.In the third chapter, the strong approximation and related embedding theorems is studied first investigated by Tikhonov. Wang already did some research in her thesis in 2010. The segmentation method here is further developed. By this method, we establish elegant results on the strong approximation and related embedding theorems under modified mean value bounded variation in real sense.The relationship between almost monotone sequences and group bounded variation sequences is quite clear, but there need be a proof. That is given in the fourth chapter by constructing the counter examples.Finally, we make a brief summary and present some problems for future works.