## Mathematics - Dissertations

#### Title

Functions of generalized bounded variation, generalized absolute continuity and applications to Fourier series

1991

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

Mathematics

Daniel Waterman

#### Keywords

Generalized bounded variation, Generalized absolute continuity, Fourier series series

Mathematics

#### Abstract

This dissertation is devoted to the study of functions of generalized bounded variation, generalized absolute continuity, and the Fourier series of such functions. We begin by defining a generalized modulus of variation which we use to define a Banach space, $\Phi$V (h), of generalized bounded variation functions which encompasses many spaces previously studied. We show that this space contains only bounded functions with simple discontinuities and that it can be written as an intersection of the Schramm spaces $\Phi$BV satisfying a certain condition. We next show that $\Phi$V (h) satisfies an analogue of Helly's theorem and that if this space contains a function which is not of harmonic bounded variation, then it does not satisfy the Dirichlet - Jordan theorem. We then show that a theorem of Zygmund may be generalized to $\Phi$V (h) as well as a larger class of generalized bounded variation spaces.

We turn our attention to Fourier series by considering the sequence $\{\alpha\sb{\rm k}$(x,f)$\}$ = $\{\rm k\ b\sb k\ cos\ kx - k\ a\sb k\ sin\ kx\}$, where a$\sb{\rm k}$ and b$\sb{\rm k}$ are the Fourier coefficients of the integrable function f. We show that this sequence is not (C,1) summable for $\Phi$V (h) functions if $\Phi$V (h) contains a function which is not of harmonic bounded variation. We show, however, that the series with terms $\alpha\sb{\rm k}$(x,f), i.e., the formally differentiated Fourier series, is (C,1) summable to (1/2) (f$\sb+\sp\prime$(x) + f$\sb-\sp\prime$(x)) if the difference quotient g(t) = (1/t) (f(x+t) $-$ f(x)) is of harmonic bounded variation in a neighborhood of t = 0. We also give conditions to insure uniform (C,1) summability of this series.

Finally, we define three generalized absolute continuity spaces,$\Lambda$AC, $\Lambda$C and $\Lambda\sb2$AC, and study the relationships between them, the space of absolutely continuous functions, and the space of continuous functions of bounded variation, BVC. We show that BVC can be written as the intersection of the $\Lambda$AC spaces and that the space of continuous functions can be written as the union of the $\Lambda$AC spaces, but that this cannot be achieved by countable intersections and unions. We end by finding a bound for the Fourier coefficients of $\Lambda\sb2$AC functions.

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