Weighted Integrability Under MVBV Condition

In analysis, trigonometric series play a very important role and has many important applications in other related sciences and engineerings. Thus, many scholars paid great attention and did some research on the convergence of trigonometric series long ago.To consider the convergence of trigonometric series, we must take its coefficients into account. For a long time, monotonicity condition setting on the coefficients were gradual-ly generalized to various bounded variation conditions, and finally, to the ultimate mean value bounded variation (MVBV) condition. Then, people are also interested in studying trigonometric integrals.Based on the series results, MVBV condition is to be generalized from sequences to functions and so weighted integrability of sine and cosine integrals under MVBV condition will be established in this thesis.The full thesis is divided into four chapters:Chapter One:IntroductionThis chapter reviews the history of the integrability problem for trigonometric series and the development are figured out briefly. Then the commonly used symbols and definitions are given.Chapter Two:Weighted integrability for MVBV functionsWang and Zhou generalized the classical Boas-Heywood theorem to the MVBV condi-tion in 2010. On this basis, we generalize weighted integrability to the MVBV functions in this chapter. The necessary and sufficient conditions are given for sine and cosine integrals of non-negative MVBV functions.Chapter Three:Weighted integrability for MVBV functions in real senseThis chapter continues the study of weighted integrability for MVBV functions, drop-ping the positivity. Through a different approach from the previous chapter, we prove:Let 0<α<1. Assume f(x)∈ MVBVF is a real function of bounded variation in [0,∞) and faa+1 xα|f(x)|dx is uniformly bounded for arbitrary a> A> 1.If then where F(t)=∫0∞ f(x) sintedx is the sine integral of f(x) on R+.A converse result is also established.Chapter four:SummaryThis chapter give a summary and an outlook in the end.