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

Approximation by translates of a single function of functions in space induced by the convolution with a given function