Galerkin information,the hyperbolic cross,and the complexity of operator equations

We obtain an exact power order of the complexity of the approximate solution of a certain class of operator equations in a Hibert space. We show that the optimal power order is realized by an algorithm that uses Galerkin information associated with the hyperbolic cross. As a corollary we derive an exact power order of the complexity of the approximate solution of Volterra integral equations whose kernels and free terms belong to Sobolev classes.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 5, pp. 639–648, May, 1991.