Let for given real sequences $(a\_n)\_{n \in \mathbb Z}, (b\_n)\_{n \in \mathbb Z}$,
$c\_n:=\sum\_{k\in \mathbb Z} a\_k b\_{n-k}$ for $n \in \mathbb Z$ be the convolution of sequences $(a\_n)$, $(b\_n)$. 


For classical convolution if one of two functions is in $L^p$, the second in $L^q$, where $1\leq p,q <\infty$ then their convolution $f\*g$ is in $L^r$, where $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}-1$.
Is it some similar type theorem true for convolution of sequences?



Convolution of sequences
Yes. True not only for $\mathbb Z$ but for for abelian (more generally unimodular) locally compact group. (20.18) in Hewitt & Ross, *Abstract Harmonic Analysis*.


