  We prove a general type description result for the multipliers acting between
two periodic Bessel potential spaces, defined on the $n$--dimensional torus, in
a case when their smoothness indices are of different signs. This is done
through the detailed examination of a periodic analogue of the linear operator
$J_s$, which is employed in the definition of the scale of the Bessel potential
space defined on the whole space $\mathbb{R}^n$. Our method of defining this
periodic analogue of $J_s$ uses the results about an asymptotic behaviour of
the generalized Fourier coefficients and existence of a natural homeomorphism
between the spaces $\mathcal{D}'(\mathbb{T}^n)$ and $S'_{2 \cdot
\pi}(\mathbb{R}^n)$, where the latter consists of all $2 \cdot \pi$--periodic
distributions from the dual Schwartz space $\mathcal{S}'(\mathbb{R}^n)$.
