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2015 Analysis of a Fractal Boundary: The Graph of the Knopp Function
Mourad Ben Slimane, Clothilde Mélot
Abstr. Appl. Anal. 2015: 1-14 (2015). DOI: 10.1155/2015/587347


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 which is defined for x0, 1 as Fx=j=02-αjϕ2jx, where 0<α<1 and ϕx=distx, z. The Knopp function itself has everywhere the same p-exponent α. 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 the 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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Mourad Ben Slimane. Clothilde Mélot. "Analysis of a Fractal Boundary: The Graph of the Knopp Function." Abstr. Appl. Anal. 2015 1 - 14, 2015.


Published: 2015
First available in Project Euclid: 15 April 2015

zbMATH: 07095581
MathSciNet: MR3305158
Digital Object Identifier: 10.1155/2015/587347

Rights: Copyright © 2015 Hindawi

Vol.2015 • 2015
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