Approximation of Hausdorff operators

Abstract

Truncating the Fourier transform averaged by means of a generalized Hausdorff operator, we approximate the adjoint to that Hausdorff operator of the given function. We find the formulas for the rate of approximation in various metrics in terms of the parameter of truncation and the components of the Hausdorff operator. Explicit rates of approximation and comparison with approximate identities are given in the case of Lipschitz α continuous functions. As an application, we show not only how to approximate the Hausdorff operator of a function, but the function itself in the L ∞ norm. After numerous works on the boundedness of Hausdorff operators on various function spaces, this paper is the first application of Hausdorff operators to the problem of constructive approximation.