#### Title

Functions Of Generalized Bounded Variation And Fourier Series

#### Date of Award

1986

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### Advisor(s)

Daniel Waterman

#### Keywords

Mathematics

#### Subject Categories

Mathematics

#### Abstract

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('(alpha)) , 0 (LESSTHEQ) (alpha) < 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.

#### Access

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#### Recommended Citation

Isaza, Pedro, "Functions Of Generalized Bounded Variation And Fourier Series" (1986). *Mathematics - Dissertations*. 51.

https://surface.syr.edu/mat_etd/51

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