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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?

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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.

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