Frame Properties of Effective Stationary Sequences

Abstract

The Kaczmarz algorithm is an iterative projection method for reconstructing vectors in a Hilbert space from inner product data. However, sequences that are effective for this algorithm need not be frames and may lack standard frame reconstruction properties. We investigate these properties for stationary effective sequences, which were classified by Kwapień and Mycielski according to their spectral measures. In particular, we show that the corresponding Kaczmarz reconstruction series always converge unconditionally if and only if the spectral measure is atomic with finitely many point masses. We show that if the spectral measure is atomic, the stationary effective sequence and its auxiliary Parseval frame can be interchanged in the Kaczmarz reconstruction formula. Finally, we construct two stationary effective sequences with singular continuous spectral measures, one of which can be interchanged with its auxiliary Parseval frame in the reconstruction formula, and one of which cannot be interchanged.