Event № 105
Given several subspaces $V_1,…,V_N$ in a Hilbert space and a point $x$, there are several algorithms for finding the orthogonal projection of $x$ on the intersection of $V_1,…,V_N$ given the orthogonal projections on each of the subspaces $V_1,…,V_N$.
I will only focus on one of these algorithms called the averaged projection method (which is a variant of the alternating projection method of von Neumann). It was shown that this algorithm either converges “very slowly” or “very quickly” and that one can assure quick convergence by bounding the (Friedrichs) angles between the subspaces (several results of this flavor were given by several authors).
In my talk, I will show how to generalize the notion of the Friedrichs angle for (linear) projections in Banach spaces in order to get a criterion for quick convergence of the averaged projection method in Banach spaces. This result does not require the projections to be of norm 1 (although the norm should be sufficiently close to 1) and therefore gives new results even in the Hilbert space setting for non-orthogonal (linear) projections.