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

摘要

We study approximation by arbitrary linear combinations of n translates of a single function of periodic functions. We construct some linear methods of this approximation for functions in a class induced by the convolution with a given function, and prove upper bounds of the Lp-approximation convergence rate by these methods, when n → ∞, for 1 < p < ∞. We also prove a lower bound 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.