Analysis of a Fractal Boundary: The Graph of the Knopp Function
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 exp...
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Wiley
2015-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2015/587347 |
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| author | Mourad Ben Slimane Clothilde Mélot |
| author_facet | Mourad Ben Slimane Clothilde Mélot |
| author_sort | Mourad Ben Slimane |
| collection | DOAJ |
| description | 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 x∈0, 1 as Fx=∑j=0∞2-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. |
| format | Article |
| id | doaj-art-c5ffcb7a5c5945d79ec91f552dcdf7f4 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2015-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-c5ffcb7a5c5945d79ec91f552dcdf7f42025-02-03T06:12:32ZengWileyAbstract and Applied Analysis1085-33751687-04092015-01-01201510.1155/2015/587347587347Analysis of a Fractal Boundary: The Graph of the Knopp FunctionMourad Ben Slimane0Clothilde Mélot1Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi ArabiaCNRS, LATP, UMR 6632, Aix-Marseille University, 13453 Marseille, FranceA 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 x∈0, 1 as Fx=∑j=0∞2-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.http://dx.doi.org/10.1155/2015/587347 |
| spellingShingle | Mourad Ben Slimane Clothilde Mélot Analysis of a Fractal Boundary: The Graph of the Knopp Function Abstract and Applied Analysis |
| title | Analysis of a Fractal Boundary: The Graph of the Knopp Function |
| title_full | Analysis of a Fractal Boundary: The Graph of the Knopp Function |
| title_fullStr | Analysis of a Fractal Boundary: The Graph of the Knopp Function |
| title_full_unstemmed | Analysis of a Fractal Boundary: The Graph of the Knopp Function |
| title_short | Analysis of a Fractal Boundary: The Graph of the Knopp Function |
| title_sort | analysis of a fractal boundary the graph of the knopp function |
| url | http://dx.doi.org/10.1155/2015/587347 |
| work_keys_str_mv | AT mouradbenslimane analysisofafractalboundarythegraphoftheknoppfunction AT clothildemelot analysisofafractalboundarythegraphoftheknoppfunction |