Fourier series with the continuous primitive integral

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 $\text{alext}$ and is a Banach space under the Alexiewicz norm, $\mathrm{\parallel }f{\mathrm{\parallel }}_{\text{T}}={\sup }_{\mathrm{\mid }I\mathrm{\mid }\le 2\pi }\mathrm{\mid }{\int }_{I}f\mathrm{\mid }$, 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 $\text{fhat}(n)=o(n)$ as $\mathrm{\mid }n\mathrm{\mid }\to \mathrm{\infty }$. The convolution is defined for $f\in \text{alext}$ and $g$ a periodic function of bounded variation. The convolution commutes with translations and is commutative and associative. There is the estimate $|f\ast g|\infty\leq |f|\T |g|\bv$\mathrm{.}For$g\in L^1(\T)$,$|f\ast g|\T\leq |f|_\T |g|_1$. As well, $\widehat{f\ast g}(n)=\text{fhatn}\widehat{g}(n)$. 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 $\text{alext}$. 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 $f\in L^1(\text{T})$. Then $\mathrm{\parallel }{D}_n\ast f-f{\mathrm{\parallel }}_{\text{T}}\to 0$ as $n\to \mathrm{\infty }$. Fourier coefficients of functions of bounded variation are characterized. An appendix contains a type of Fubini theorem.