  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.

Classical Analysis and ODEs