# A Kaczmarz algorithm for sequences of projections, infinite products, and applications to frames in IFS $L^{2}$ spaces

## TL;DR

This paper extends the Kaczmarz algorithm to infinite-dimensional, non-commutative settings, demonstrating its applications in spectral theory, optimization, IFS, and fractal harmonic analysis, with new recursive schemes and error estimates.

## Contribution

It introduces a novel recursive iteration scheme for sequences of projections, expanding the Kaczmarz algorithm's applicability to complex infinite-dimensional and non-commutative contexts.

## Key findings

- Development of a new recursive iteration scheme for selfadjoint projections
- Applications to random Kaczmarz recursions and their limits
- Error estimates for the proposed recursive schemes

## Abstract

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.