Functions Of Generalized Bounded Variation And Fourier Series

This dissertation is devoted to the study of some properties and applications of functions of generalized bounded variation. Estimates are obtained for the Fourier coefficients of a function f whose Fourier series has small gaps and whose restriction to a subinterval I of 0,2(pi) , f(VBAR)(,I), belongs to one of the following classes: (PHI) bounded variation, (WEDGE) bounded variation, or V h of Canturija. A condition is obtained for the absolute convergence of the Fourier series of f when f(VBAR)(,I) is in V n(\u27(alpha)) , 0 (LESSTHEQ) (alpha) \u3c 1/2. Hypotheses for the existence of the Stieltjes integral of functions in Canturija classes are given and the integral is estimated. It is proved that each space V h is the intersection of all (WEDGE)B(V) classes satisfying certain conditions, but is not the intersection of any countable subcollection of these classes. Finally, a definition is given for a Banach space of regulated functions in a manner analogous to that for functions of ordered harmonic bounded variation, but using only intervals of equal length and requiring that the functions satisfy a generalized continuity condition. It is shown that functions in this space have everywhere convergent Fourier series