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

摘要

We study approximation properties of the sums Sigma(n)(k=1) f(t-a(k)) of shifts of a single function f in real spaces L-p(T) and C(T) on the circle 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(T) and obtain bounds for the rate of approximation.