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Analysis of a fractal boundary: the graph of the Knopp function

Abstract : A usual classification tool to study a fractal interface is the computation of its fractal dimension. But a recent method developed by Y. Heurteaux and S. Jaffard proposes to compute either weak and strong accessibility exponents or local Lp regularity exponents (the so-called p-exponent). These exponents describe locally the behavior of the interface. We apply this method to the graph of the Knopp function. The Knopp function itself has everywhere the same p-exponent. Nevertheless, using the characterization of the maxima and minima done by B. Dubuc and S. Dubuc, we will compute the p-exponent of the characteristic function of domain under the graph of F at each point (x,F(x)) and show that p-exponents, weak and strong accessibility exponents change from point to point. Furthermore we will derive a characterization of the local extrema of the function according to the values of these exponents.
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Submitted on : Wednesday, July 23, 2014 - 3:02:49 AM
Last modification on : Friday, October 22, 2021 - 3:31:28 AM
Long-term archiving on: : Tuesday, November 25, 2014 - 11:46:19 AM


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  • HAL Id : hal-01037859, version 1
  • ARXIV : 1407.6219



Mourad Ben Slimane, Clothilde Melot. Analysis of a fractal boundary: the graph of the Knopp function. Abstract and Applied Analysis, 2015, ⟨hal-01037859⟩



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