Analysis of a Fractal Boundary: The Graph of the Knopp Function. (22nd January 2015)

• Record Type: Journal Article
• Authors: Ben Slimane, Mourad; Mélot, Clothilde
• Other Names: Zhang Weinian Academic Editor.
• Is Part Of: Abstract and applied analysis. Volume 2015(2015)
• Publication Date: 2015-01-22
• Subjects: Mathematical analysis -- Periodicals; Mathematical analysis; Applied Mathematics; Mathematical Analysis; Periodicals; 515.05
• DOI: 10.1155/2015/587347
• Languages: English
• ISSNs: 1085-3375

Abstract

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 L p 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 x ∈ 0, 1 as F x = ∑ j = 0 ∞ 2 - α j ϕ 2 j x, where 0 < α < 1 and ϕ x = dist x, 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.