Determination Of Jumps For Functions

The thesis includes three Chapters.The first Chapter is an introduction.In Chapter 2 we discussed concentration factor method for determination of jumps of periodic functions at their simple discontinuities. General concentration factor method was introduced by Gelb and Tadmor[5] in 1999. They established a criterion on concentration factors. Later, Q. L. Shi and X. L. Shi[6] proved an improvement. In these results the functions are assumed to satisfy a weak smoothness condition. Via special concentration factors Golubov[2] and Kvernadze[4] established convergence theorems on determination of jumps for functions in the class V_p, (p≥1) and class HBV, respectively. In this Chapter we considered functions in more general class HBMV and more general concentration factors. Our theorem improved results obtained by Golubov and Kvernadze. This part will be published on the journal "Acta Math. Hungar."In Chapter 3 we discussed determination of jumps for non-periodic functions. We proved the equality of Lukacs type for Malvar-Coifman-Meyer conjugate wavelets. Furthermore we established several criteria on concentration factors for functions that satisfy weak-smoothness condition of Dini type. The Hilbert transform of MCM wavelet has no obvious expression. By the study of Hilbert commutators we solved this difficulty and established several criteria on concentration factors. This part was also submitted to a journal of SCI.