Galerkin information, the hyperbolic cross, and the complexity of operator equations (Q1177764)

Let \(H\) be a Hilbert space and let \(X\) be a continuous embedded normal subspace of \(H\). The author studies the convergence of an iterative process which abouts to a solution of the equation: \[ x=Ax+b, \] where \(b\in X\) and \(A\) is a linear operator on \(H\) which is ``smooth'' with respect to the space \(X\). More exactly \(A\) and \(A^*\) are bounded operators from \(H\) into \(X\). Optimal estimates for the arts of convergence and the complexity of the computations involved in the respective iterative process are amply discussed.