On Spherical Convergence, Convexity, and Block Iterative Projection Algorithms in Hilbert Space

We call a sequence xn4 in Hilbert space ‘‘spherical’’ if there exists u such that lim5xnyu5exists and is finite. If moreover u is a weak accumulation point of the sequence, we call the sequence ‘‘spherically convergent.’’ We demonstrate that for large classes of nonexpansive possibly nonstationary. discrete-time processes in Hilbert space the iterates are spherically convergent. Basic identities and orthogonality relations pertinent to this type of convergence are exhibited. Sufficient conditions for weak and spherical convergence, in terms of the ‘‘fullness’’ of the set of fixed points of the process, are established and compared. Spherical convergence of the general block iterative projection scheme in Hilbert space is established.