| Abstract: | In Kolwankar and Lévy Véhel, new functional spaces, denoted $K^{s,s }_{x_0}$, wereintroduced. These spaces characterize the fine local regularity offunctions, much in the spirit of 2-microlocal spaces$C^{s,s}_{x_0}$. In contrast with $C^{s,s}_{x_0}$ spaces, however, $K^{s,s}_{x_0}$spaces are defined through simple estimations on the pointwise valuesof the functions. In this work, we generalize the definition of$K^{s,s}_{x_0}$ spaces and prove the equality $C^{s,s}_{x_0}=K^{s,s}_{x_0}$ for $s+s>0$, $s>0$.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  difficult functions suchas continuous but nowhere differentiable ones. As a by-product, robustestimators of both the pointwise and the local exponents are obtained. |
