Some Extensions Of A Class Of Asymptotic Formulae In Analysis

The theory of trigonometric series is a vast field in mathematics, and trigonometric series certainly contain Fourier series as an elementary concept in Fourier Analysis. Under monotonicity and positivity condition, Chaundy and Jolliffe proved the uniform conver-gence of sine series. After that, people gradually generalized monotonicity to various quasi-monotonicity, such as quasi-monotonicity, regular-varying quasi-monotonicity condition and O-regular-varying quasi-monotonicity.In2001, Hungarian mathematician Leindler shifted his attention to the Rest Bounded Variation (RBV) concept to generalize monotonicity. In another direction, he also found (in2002) that RBV Sequences and QM Sequences cannot contain each other. In2005, Le and Zhou raised Group Bounded Variation concept to contain RBV concept and QM concept and various quasi-monotonicity. Finally, Zhou etc. gave the concept of Mean Value Bounded Variation(MVBV) in2010. Plenty of classical results such as the uniform convergence of sine and cosine series, L1-convergence of Fourier series, Lp-integrability and so on are generalized to MVBV condition.The asymptotic formula of sine and cosine series was proved in Zygmund’s book,"trigonometric series", and was generalized by Hardy under monotonicity with a necessity and sufficiencient condition given there. After that, some other extensions were developed. In1992, Nurcomb proved this asymptotic formula in quasi-monotonicity. It is interesting that Xie and Zhou in1994pointed out that, the sufficiency of the formula cannot hold in RQM condition, while the necessity can be extended. Later Le, Zhou, Wang and Zhao generalized the asymptotic formula in GBV and MVBV conditions, they also proved the L2π-integrability.The first object of this article is to establish the relationship among these conditions, motivatived by the idea in Leindler[8]. We know that, Fourier transformation plays an im-portant role in computation and engineering. The second object of the article is to establish corresponding results in Fourier transformation.The full article is divided into four chapters:The first chapter gives some relative backgrounds and works that already done on these topics. Some definitions are also listed here. In the second chapter, from Le and Zhou’s theorem and Leindler’s work, we deduced an equivalent relation between GBV sequenses and MVBV sequences (in generalized form) under condition Moreover, we construct a counterexample to show that the equivalence cannot hold without condition (1.2). In addition, we also study if there is equivalence among GBV sequences, RQM sequences and QM sequences.In the third chapter, we considered MVBV functions and obtained some asymptotic formulae for Fourier transform, and proved the equivalence between MVBV functions and RQM functions.In the fourth chapter, we generalized a useful lemma in Chapter2, and an elaborate example is given to show that the MVBV condition cannot be dropped in Theorem4.2.1.