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Old wine in fractal bottles I: Orthogonal expansions on self-referential spaces via fractal transformations

Author

Listed:
  • Bandt, Christoph
  • Barnsley, Michael
  • Hegland, Markus
  • Vince, Andrew

Abstract

Our results and examples show how transformations between self-similar sets may be continuous almost everywhere with respect to measures on the sets and may be used to carry well known notions from analysis and functional analysis, for example flows and spectral analysis, from familiar settings to new ones. The focus of this paper is on a number of surprising applications including what we call fractal Fourier analysis, in which the graphs of the basis functions are Cantor sets, discontinuous at a countable dense set of points, yet have good approximation properties. In a sequel, the focus will be on Lebesgue measure-preserving flows whose wave-fronts are fractals. The key idea is to use fractal transformations to provide unitary transformations between Hilbert spaces defined on attractors of iterated function systems.

Suggested Citation

  • Bandt, Christoph & Barnsley, Michael & Hegland, Markus & Vince, Andrew, 2016. "Old wine in fractal bottles I: Orthogonal expansions on self-referential spaces via fractal transformations," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 478-489.
    • Handle: RePEc:eee:chsofr:v:91:y:2016:i:c:p:478-489
    DOI: 10.1016/j.chaos.2016.07.007
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    Cited by:

    1. Sánchez-Granero, M.A. & Fernández-Martínez, M., 2019. "Irreducible fractal structures for Moran type theorems," Chaos, Solitons & Fractals, Elsevier, vol. 119(C), pages 29-36.
    2. Massopust, Peter R., 2022. "Fractal interpolation over nonlinear partitions," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).
    3. Barnsley, Louisa F. & Barnsley, Michael F. & Vince, Andrew, 2024. "Tiling iterated function systems," Chaos, Solitons & Fractals, Elsevier, vol. 182(C).

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