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 M-i, i = 1, . . ., m, of a real Hilbert space H. We show that the average distance to individual sets enjoys a polynomial behavior o(k(-1/2)) along the trajectory of the generated iterates. Surprisingly, when the starting points are chosen from the subspace S-i=1(m) M-i(?), our result yields a polynomial rate of convergence O(k(-1/2)) for the method of cyclic projections itself. Moreover, if E(i=1)(m)M(i)(?)is not closed, then both of the aforementioned rates are best possible in the sense that the corresponding polynomial k(1/2) cannot be replaced by k(1/2+e) for any e > 0.