Estimates of efficiency for two methods of stable numerical summation of smooth functions

Abstract

We consider a classical ill-posed problem of reconstruction of continuous functions from their noisy Fourier coefficients. We study the case of functions of two variables that has been much less investigated. The smoothness of reconstructed functions is measured in terms of the Sobolev classes as well as the classes of functions with dominated mixed derivatives. We investigate two summation methods, that are based on ideas of the rectangle and the hyperbolic cross, respectively. For both of these methods we establish the estimates of the accuracy on the classes that are considered as well the estimates of computational costs. Moreover, we made the comparison of their efficiency based on obtained estimates. A somehow surprising outcome of our study is that for both types of the considered smoothness classes one should employ hyperbolic cross approximation that is not typical for the functions under consideration.