We consider the smoothness parameter of a function f ∈ L²(ℝ) in terms of Besov spaces , . The existing results on estimation of smoothness [K. Dziedziul, M. Kucharska and B. Wolnik, J. Nonparametric Statist. 23 (2011)] employ the Haar basis and are limited to the case 0 < s*(f) < 1/2. Using p-regular (p ≥ 1) spline wavelets with exponential decay we extend them to density functions with 0 < s*(f) < p+1/2. Applying the Franklin-Strömberg wavelet p = 1, we prove that the presented estimator of s*(f) is consistent for piecewise constant functions. Furthermore, we show that the results for the Franklin-Strömberg wavelet can be generalised to any spline wavelet (p ≥ 1).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-am40-3-4,
author = {Magdalena Meller and Natalia Jarz\k ebkowska},
title = {Estimation of a smoothness parameter by spline wavelets},
journal = {Applicationes Mathematicae},
volume = {40},
year = {2013},
pages = {309-326},
zbl = {06238620},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-am40-3-4}
}
Magdalena Meller; Natalia Jarzębkowska. Estimation of a smoothness parameter by spline wavelets. Applicationes Mathematicae, Tome 40 (2013) pp. 309-326. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-am40-3-4/