  We connect through the Fourier transform shift-invariant Sobolev type spaces
$V_s\subset H^s$, $s\in\mathbb R,$ and the spaces of periodic distributions and
analyze the properties of elements in such spaces with respect to the product.
If the series expansions of two periodic distributions have compatible
coefficient estimates, then their product is a periodic tempered distribution.
We connect product of tempered distributions with the product of
shift-invariant elements of $V_s$. The idea for the analysis of products comes
from the H\"ormander's description of the Sobolev type wave front in connection
with the product of distributions. Coefficient compatibility for the product of
$f$ and $g$ in the case of "good" position of their Sobolev type wave fronts is
proved in the 2-dimensional case. For larger dimension it is an open problem
because of the difficulties on the description of the intersection of cones in
dimension $d\geqslant3$.
