A Kaczmarz algorithm for sequences of projections, infinite products, and applications to frames in IFS $L^{2}$ spaces
This work addresses the need for iterative projection methods in infinite-dimensional and non-commutative contexts, which is relevant to spectral theory, optimization, and fractal analysis.
The paper extends the Kaczmarz algorithm to infinite-dimensional, non-commutative settings involving sequences of selfadjoint projections, with applications to iterated function systems and fractal harmonic analysis. It provides new recursive schemes and error estimates for random Kaczmarz recursions.
We show that an idea, originating initially with a fundamental recursive iteration scheme (usually referred as "the" Kaczmarz algorithm), admits important applications in such infinite-dimensional, and non-commutative, settings as are central to spectral theory of operators in Hilbert space, to optimization, to large sparse systems, to iterated function systems (IFS), and to fractal harmonic analysis. We present a new recursive iteration scheme involving as input a prescribed sequence of selfadjoint projections. Applications include random Kaczmarz recursions, their limits, and their error-estimates.