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A Time DomainCharacterization of2-Microlocal Spaces
Authors:Stéphane?Seuret  author-information"  >  author-information__contact u-icon-before"  >  mailto:stephane.seuret@inria.fr"   title="  stephane.seuret@inria.fr"   itemprop="  email"   data-track="  click"   data-track-action="  Email author"   data-track-label="  "  >Email author,Jacques?Lévy Véhel  author-information"  >  author-information__contact u-icon-before"  >  mailto:jacques.levy_vehel@inria.fr"   title="  jacques.levy_vehel@inria.fr"   itemprop="  email"   data-track="  click"   data-track-action="  Email author"   data-track-label="  "  >Email author
Affiliation:(1) Projet Fractales, Inria Rocquencourt, B.P. 105, 78153 Le Chesnay Cedex, France
Abstract:In Kolwankar and Lévy Véhel, new functional spaces, denoted $K^{s,srsquo}_{x_0}$, wereintroduced. These spaces characterize the fine local regularity offunctions, much in the spirit of 2-microlocal spaces$C^{s,srsquo}_{x_0}$. In contrast with $C^{s,srsquo}_{x_0}$ spaces, however, $K^{s,srsquo}_{x_0}$spaces are defined through simple estimations on the pointwise valuesof the functions. In this work, we generalize the definition of$K^{s,srsquo}_{x_0}$ spaces and prove the equality $C^{s,srsquo}_{x_0}=K^{s,srsquo}_{x_0}$ for $s+srsquo>0$, $s>0$.Using this result, we propose an algorithm able to estimate a part ofthe 2-microlocal frontier. Experiments on sampled data show thatreasonable accuracy is achieved even for ldquodifficultrdquo functions suchas continuous but nowhere differentiable ones. As a by-product, robustestimators of both the pointwise and the local exponents are obtained.
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