Methods of solving Fredholm equations optimal on classes of functions

This paper is devoted to the solution of linear Fredholm equations in the unit s-dimensional cube for classes of functions with a dominant mixed derivative of order r in each variable. We present an algorithm for obtaining the solution over the whole domain with an error O(N^?r ln^2s?1 N) in the uniform metric using the values of the given functions at O(N ln^2s?1 N) points and consisting of O(N ln^2s?1 N) elementary operations. We show that these estimates can only be improved at the expense of the exponent of ln N.