Fourier series with the continuous primitive integral

Abstract

Fourier series are considered on the one-dimensional torus for the space of periodic distributions that are the distributional derivative of a continuous function. This space of distributions is denoted $\alext$ and is a Banach space under the Alexiewicz norm, $\|f\|_\T =\sup_{|I|\leq 2\pi}|\int_I f|$, the supremum being taken over intervals of length not exceeding $2\pi$. It contains the periodic functions integrable in the sense of Lebesgue and Henstock-Kurzweil. Many of the properties of $L^1$ Fourier series continue to hold for this larger space, with the $L^1$ norm replaced by the Alexiewicz norm. The Riemann-Lebesgue lemma takes the form $\fhat(n)=o(n)$ as $|n|\to\infty$. The convolution is defined for $f\in\alext$ and $g$ a periodic function of bounded variation. The convolution commutes with translations and is commutative and associative. There is the estimate $\|f\ast g\|_\infty\leq \|f\|_\T \|g\|_\bv$. For $g\in L^1(\T)$, $\|f\ast g\|_\T\leq \|f\|_\T \|g\|_1$. As well, $\widehat{f\ast g}(n)=\fhatn \hat{g}(n)$. There are versions of the Salem-Zygmund-Rudin-Cohen factorization theorem, Fej\'er's lemma and the Parseval equality. The trigonometric polynomials are dense in $\alext$. The convolution of $f$ with a sequence of summability kernels converges to $f$ in the Alexiewicz norm. Let $D_n$ be the Dirichlet kernel and let $f\in L^1(\T)$. Then $\|D_n\ast f-f\|_\T\to 0$ as $n\to\infty$. Fourier coefficients of functions of bounded variation are characterized. An appendix contains a type of Fubini theorem.