On the approximation by convolution operators in homogeneous Banach spaces on R^d

AMS Subject Classification 2010: 41A25, 41A35, 41A40, 41A63, 41A65, 42A38, 42A85, 42B10, 42B20

The paper presents a description of the optimal rate of approximation as well as of a broad class of functions that possess it for convolution operators acting in the so-called homogeneous Banach spaces of functions on Rd. The description is the same in any such space and uses the Fourier transform. Simple criteria for establishing upper estimates of the approximation error via a K-functional are given. The differential operator in the K-functional is defined similarly to the infinitesimal generator by means of the convolution operator.