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

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.

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Acknowledgements

The co-authors thank the following colleagues for helpful and enlightening discussions: Professors Daniel Alpay, Sergii Bezuglyi, Ilwoo Cho, Wayne Polyzou, Eric S. Weber, and members in the Math Physics seminar at The University of Iowa.

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Correspondence to James Tian.

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Communicated by Uwe Franz.

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Jorgensen, P., Song, M. & Tian, J. A Kaczmarz algorithm for sequences of projections, infinite products, and applications to frames in IFS \(L^{2}\) spaces. Adv. Oper. Theory (2020). https://doi.org/10.1007/s43036-020-00079-1

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Keywords

  • Hilbert space
  • Frames
  • Kaczmarz algorithm
  • Randomized Kaczmarz algorithm
  • Sierpinski gasket
  • Infinite products
  • Analysis/synthesis
  • Iterated function system

Mathematics Subject Classification

  • 47L60
  • 46N30
  • 46N50
  • 42C15
  • 65R10
  • 60J20