Say $f\in L^1({\bf T}^n)$ a function on the real n-torus ${\bf T}^n$ with $\sum_{{\bf m}\in {\bf Z}^n} \mid \hat{f}({\bf m})\mid<\infty$ where $\hat{f}({\bf m})=\int_{{\bf T}^n} f({\bf x}) e^{-2\pi i {\bf m}\cdot {\bf x}} dx$ is the m-th Fourier coefficient. Then by Fourier inversion
$f({\bf x})= \sum_{{\bf m}\in {\bf Z}^n}\hat{f}({\bf m})e^{2\pi i {\bf m}\cdot {\bf x}}$
almost everywhere. Now set $\Delta: {\bf Z}\rightarrow {\bf Z}^n, m \mapsto (m,\ldots,m)^t$. In this setting what (if anything non-trivial) can one say about
$\sum_{{\bf m}\in \Delta({\bf Z})}\hat{f}({\bf m})e^{2\pi i {\bf m}\cdot {\bf x}}$
or
$\sum_{{\bf m}\in \Delta({\bf Z})} \mid \hat{f}({\bf m})\mid$?
What I mean is, are there theorems (with perhaps additional assumptions) making any interesting statements about these sums?