Fourier series on Vilenkin groups
Date of Award
Doctor of Philosophy (PhD)
Boundedness, Bounded sequence, Salem test
The focus of this investigation is pointwise convergence of Fourier series of functions defined on a compact Vilenkin group, i.e., a compact metrizable totally disconnected abelian group. Throughout we assume that the Vilenkin group satisfies a certain boundedness condition, namely, that a sequence of primes determined by the structure of the group is a bounded sequence.
We begin with an examination of the Salem test for uniform convergence of Fourier series. We provide a new proof of an adaptation of the Salem test to Fourier series on Vilenkin groups due to C. W. Onneweer and Daniel Waterman.
Next we localize the Salem test to obtain a test for pointwise convergence of the Fourier series of f. Here it is shown that the assumption of continuity of f at x, which was required in the proof of uniform convergence, may be weakened to the existence of a suitably defined derivative of the integral of f(x-t). This localized Salem test is very closely related to a version of the Lebesgue test due to Onneweer and Waterman.
Application of similar methods to the study of functions of harmonic bounded fluctuation yields the result that the Fourier series of a function of harmonic bounded fluctuation converges everywhere the function satisfies the differentiability of the integral condition mentioned above.
Finally, we construct a Banach space of functions with everywhere-convergent Fourier series.
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Dezern, David Herman, "Fourier series on Vilenkin groups" (1988). Mathematics - Dissertations. 54.