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

Mourad Ben Slimane (1) , Clothilde Melot (2)

1 Department of Mathematics [Riyadh]
2 I2M - Institut de Mathématiques de Marseille

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.

Mots clés: Extrema; Fractal interface; Knopp function; H ̈older and Lp regularities; Weak and strong accessibility exponents; Dyadic expansion; Extrema.

Domaines: Analyse classique [math.CA]; Mathématiques générales [math.GM]; Traitement du signal et de l'image [eess.SP]

Mourad Ben Slimane, Clothilde Melot. Analysis of a fractal boundary: the graph of the Knopp function. Abstract and Applied Analysis, 2015, http://www.hindawi.com/journals/aaa/2015/587347/. ⟨hal-01037859⟩