The Best Approximation And Convergence Of Sine Series

The main content of this thesis is the best approximation and convergence of sine series under different condition. Many classical results in Fourier analysis are achieved by assuming the monotonicity of the coefficient sequence of trigonometric series. Especially, Chaudy and Jollife research convergence uniformly of sine series if the coefficient is a nonnegative and decreasing sequence. Very recently, people introduce strong mean value bound variation sequence (denote MVBVS*) and piecewise bounded variation, we will get the relation among the best approximation and coefficients of trigonometric series. Finally combining with logarithm bounded variation condition and Moricz theorem we get a new result.This thesis can be divided into five chapters:The first chapter is introduction. We introduced the background of the research contents and the status of current studies. We also will give some symbols and notations frequently used.The second chapter is the relations among the best approximation and coefficients of sine series under the condition of strong mean value bounded variation. At first, L.Leindler given the condition of rest bounded variation and got the relationship between the coefficient of trigonometric series and best approximation of sine series. With the development of the monotonicity conditions to the condition of non-onesided bounded variation, Mei Y.-Wei B.R. has been generalized the theorem and get the same result. In the end, under the condition of the mean value bounded variation, people also research this, but the result is not perfect as the definition of mean value bounded variation. In C space(all functions continuous) it has a similar trouble. Then people get the strong mean bounded variation weaker than the mean bounded variation, and solve the problem. In this paper, we will use this condition generalized the L.Leindler theorem.Our content is that the coefficients must be nonnegative. Then people consider whether the nonnegative condition can be cancelled? If not, what the condition can replace. Zhou got piecewise bounded variation condition. Some people use this condition have gotten L-convergence of Fourier series. The third chapter, we research the convergence of best approximation of sine series, and get the relationship between best approximation and coef-ficient.The fourth chapter generalized Moricz theorem. In studying L-convergence problems, people usually need a requirement that the g∈L, which also become a hard condition to check or a prior condition to set in most cases. We know that the prior condition in L1case cannot be avoided mainly arises from the much more "computation complexity" in the integrable space than the continuous space. So people would prefer not integrable conditions of subject research. The initial results is Boas-Hoywood’s monotonicity result, then Moricz has been generalized it to double series. Recently, Zhou get logarithm bounded variation condition and has made breakthrough progress. Based on the logarithm bounded variation conditions, we has been generalized Moricz theorem to double series using different method.Finally, we summarize and discuss some problems for future research.