Function approximation by singular integral and applications

Abstract

Function approximation by convolution type singular integrals has important applications in differential and integral equations. In this paper we study general singular operators. We first develop the test conditions for the convergence of convolution type singular integral operators to approximated function in the exponential weighted space. Then we propose, for estimating the rate of approximation, in new modulus of smoothness and examine the main properties of this modulus of smoothness. We also give some applications for the Gauss-Weierstrass integral.