Abstract:
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.
Keywords:approximation, sums of shifts, Fourier coefficients, semigroup.