Approximation by sums of shifts of a single function on the circle

We study approximation properties of the sums $\sum_{k=1}^nf(t-a_k)$ of shifts of a single function $f$ in real spaces $L_p(\mathbb{T})$ and $C(\mathbb{T})$ on the circle $\mathbb{T}=[0,2\pi)$, and also in complex spaces of functions analytic in the unit disc. We obtain sufficient conditions in terms of the trigonometric Fourier coefficients of $f$ for these sums to be dense in the corresponding subspaces of functions with zero mean. We investigate the accuracy of these conditions. We also suggest a simple algorithm for the approximation by sums of plus or minus shifts of one particular function in $L_2(\mathbb{T})$ and obtain bounds for the rate of approximation.