Construction of Continuous Functions with Prescribed Local Regularity |
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Authors: | K Daoudi J Lévy Véhel Y Meyer |
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Institution: | (1) INRIA Rocquencourt BP 105 78153 Le Chesnay Cedex France daoudi@bora.inria.fr, FR;(2) INRIA Rocquencourt BP 105 78153 Le Chesnay Cedex France jlv@bora.inria.fr, FR;(3) Y. Meyer Université Paris IX Dauphine CEREMADE Place du Maréchal de Lattre De Tassigny 75775 Paris France meyer@ceremade.dauphine.fr, FR |
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Abstract: | In this paper we investigate from both a theoretical and a practical point of view the following problem: Let s be a function from 0;1] to 0;1] . Under which conditions does there exist a continuous function f from 0;1] to R such that the regularity of f at x , measured in terms of H?lder exponent, is exactly s(x) , for all x ∈ 0;1] ?
We obtain a necessary and sufficient condition on s and give three constructions of the associated function f . We also examine some extensions regarding, for instance, the box or Tricot dimension or the multifractal spectrum. Finally,
we present a result on the ``size' of the set of functions with prescribed local regularity.
November 30, 1995. Date revised: September 30, 1996. |
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Keywords: | , H?lder exponents, Weierstrass functions, Schauder basis, IFS, Fractals, AMS Classification, 28A80, |
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