Title

General properties of functions of bounded lambda-variation

Date of Award

1995

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Advisor(s)

Daniel Waterman

Keywords

lambda variation, bounded lambda variation, Garsia Sawyer class

Subject Categories

Mathematics

Abstract

This thesis is focused on general properties of functions of bounded $\lambda$-variation. Inspiration for most of this work came from problems posed by D. Waterman, and four of them have been answered here completely. It has been shown that functions of bounded ordered $\lambda$-variation always form a proper superset of the class of functions of bounded $\lambda$-variation. A useful characterization of functions continuous in $\lambda$-variation has been found, a characterization that generalizes the decomposition of ordinary variation of a function into the sum of the variation of the continuous part of the function and the variation of saltus part of the function. A necessary and sufficient condition for the equivalence of the concepts of bounded $\lambda$-variation and continuity in $\lambda$-variation has been proven. A notion of $\lambda$-absolute continuity has been introduced, and various interesting characterizations of it has been given. Finally, all the previous results interweave beautifully in an explanation of the relationship between harmonic bounded variation and the Garsia-Sawyer class.

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