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.