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Harmonic analysis on fractal spaces

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Harmonic analysis on fractal spaces SÉMINAIRE N. BOURBAKI MARTINBARLOW Harmonic analysis on fractal spaces Séminaire N. Bourbaki, 1991-1992, exp. no 755, p. 345-368. <> © Association des collaborateurs de Nicolas Bourbaki, 1991-1992, tous droits réservés. L’accès aux archives du séminaire Bourbaki (http://www.bourbaki. implique l’accord avec les conditions générales d’utilisa- tion ( Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques 345 HARMONIC ANALYSIS ON FRACTAL SPACES par Martin BARLOW Seminaire BOURBAKI 44ème ann~e, 1991-92, n° 755 Juin 1992 1. INTRODUCTION The initial interest in this area came from mathematical physicists in the early 1980s, who were studying transport properties of disordered media. As there are good reasons for believing that fractals can provide a good model for such media, this led to their interest in such questions on the study of the wave and heat equations on fractal spaces - see for example [RT]. Mathematical work has, so far, largely centered on the more easily treated Laplace and heat equations. It began with a probabilistic treat- ment, by Kusuoka [Kl], Goldstein [G] and Barlow-Perkins [BP], but an analytic treatment, making a substantial use of Dirichlet forms has been developed, mainly in Japan - see [Kigl, Kig2, F2, K2]. Most of the problems and difficulties arise, naturally, on the micro- scopic scale, from absence of any kind of Euclidean structure. However, given a regular fractal, such as the Sierpinski gasket, one can define a pre-fractal manifold or graph, whose large scale structure mimics that of the true fractal. These pre-fractals are entirely classical, but classi

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