Polynomial Estimates for the Method of Cyclic Projections in Hilbert Spaces

Abstract

We study the method of cyclic projections when applied to closed and linear subspaces $M_i$, $i=1,\dots ,m$, 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 ${\sum }_{i=1}^m{M}_i^{\perp }$, our result yields a polynomial rate of convergence $\mathcal{O}(k^{-1/2})$ for the method of cyclic projections itself. Moreover, if ${\sum }_{i=1}^m{M}_i^{\perp }$ 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 $k^{1/2+\epsilon }$ for any $\epsilon >0$.