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In chap. 3 of "Fractal Geometry of Nature" Mandelbrot mentions that "part of the study of fractals is the geometric face of harmonic analysis" (spectral or Fourier, he specifies), but to my dismay, this is immediately followed by "...but this fact is not stressed in the present work."

Google hasn't turned anything up. Can anyone here orient me on how fractals and harmonic analysis are related?

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I can only guess at the connection. Harmonic analysis is concerned with investigating the behavior of functions localized at different scales. Classically, this localization is realized by examining frequency space. Similarly, fractals are concerned with behavior at varying scales (in particular, self-similarity at many scales). – Christopher A. Wong Oct 11 '12 at 6:11
Look for the work of Fefferman on the problem of the ball as a multiplier, this is one of the most celebrated connections between harmonic analysis and fractal geometry. Also google for Kakeya problem. – user39490 Jul 19 '14 at 1:12

Just to list a few connections:

  • Fefferman's $L^p$ multiplier counterexample and Kakeya sets.

See "Modern Fourier Analysis" by Loukas Grafakos

  • The construction and analysis of the Laplacian on the sierpinski gasket.

See "Energy and Laplacian on the Sierpinksi Gasket" by Alexander Teplyaev

  • The spectral decimation of eigenvalues

See "Analysis on Fractals" by Robert Strichartz

  • Schrödinger operators and fractal potentials

See the paper "On Schrödinger Operators Perturbed by Fractal Potentials" by Sergio Albeverio

  • The study of pseudo-differential operators on fractals

See "Fractals and Spectra" by Hans Triebel, and "PDOs on Fractals" by Marius Ionescu, Luke Rogers and Robert Strichartz

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