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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

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