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Pré-Publication, Document De Travail Année : 2017

Laplacian, on the graph of the Weierstrass-Hadamard function

Claire David

Résumé

Following our Laplacian on the graph of the Weierstrass function, it was natural to consider, then, the case of the Weierstrass-Hadamard function (lacunary complex series). The novelty consists in working into an entirely complex space. We present thus, in the following, the results obtained by following the approach of J. Kigami and R. S. Strichartz. Ours is made in a completely renewed framework, as regards, the one, affine, of the Sierpinski gasket. First, we concentrate on Dirichlet forms, on the graph of the Weierstrass function, which enable us the, subject to its existence, to define the Laplacian of a continuous function on this graph. This Laplacian appears as the renormalized limit of a sequence of discrete Laplacians on a sequence of graphs which converge to the one of the Weierstrass function. The normalization constants related to each graph Laplacian are obtained thanks Dirichlet forms. In addition to the Dirichlet forms, we have come across several delicate points: the building of a self-similar measure related to the graph of the function, as well as the one of spline functions on the vertices of the graph. The spectrum of the Laplacian thus built is obtained through spectral decimation. Beautifully, as regards to the method developed by Robert S. Strichartz in the case of the de Sierpinski gasket, our results come as the most natural illustration of the iterative process that gives birth to the discrete sequence of graphs.
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Dates et versions

hal-01487602 , version 1 (14-03-2017)

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  • HAL Id : hal-01487602 , version 1

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Claire David. Laplacian, on the graph of the Weierstrass-Hadamard function. 2017. ⟨hal-01487602⟩
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