  We study approximation by arbitrary linear combinations of $n$ translates of
a single function of periodic functions. We construct some methods of this
approximation for functions in a class induced by the convolution with a given
function, and prove upper bounds of $L_p$-the approximation convergence rate by
these methods, when $n \to \infty$, for $1 < p < \infty$, and lower bounds of
the quantity of best approximation of this class by arbitrary linear
combinations of $n$ translates of arbitrary function, for the particular case
$p=2$.
