Polynomial Estimates for the Method of Cyclic Projections in Hilbert Spaces

We study the method of cyclic projections when applied to closed and linear subspaces Mi, i=1,ldots,m, of a real Hilbert space mathcal H. We show that the average distance to individual sets enjoys a polynomial behaviour o(k-1/2) along the trajectory of the generated iterates. Surprisingly, when the starting points are chosen from the subspace sumi=1mMi^perp, our result yields a polynomial rate of convergence mathcal O(k-1/2) for the method of cyclic projections itself. Moreover, if sumi=1m Mi^perp is not closed, then both of the aforementioned rates are best possible in the sense that the corresponding polynomial k1/2 cannot be replaced by k1/2+varepsilon for any varepsilon >0.