Complexity of Local Solution of Multivariate Integral Equations

In this paper the complexity of the local solution of Fredholm integral equations is studied. For certain Sobolev classes of multivariate periodic functions with dominating mixed derivative we prove matching lower and upper hounds. The lower bound is shown using relations to s-numbers. The upper hound is proved in a constructive way providing an implementable algorithm of optimal order based on Fourier coefficients and a hyperbolic cross approximation.