This paper represents the text of a minicourse delivered by the author
for participants of the \textquotedbl{}Workshop on Measure Theory
and Real Analysis\textquotedbl{}, Grado, September-97. We consider
topics on multiplier theory and translation-invariant subspaces in
function spaces on groups $\mathbb{T}$ and $\mathbb{Z}$. Our goal
is to give an introduction to the subject from the very beginning
up to some recent results. The presentation is aimed at graduate students;
no preliminaries in Fourier Analysis are supposed.
