Estimation of a smoothness parameter by spline wavelets

Magdalena Meller, and Natalia Jarzębkowska. "Estimation of a smoothness parameter by spline wavelets." Applicationes Mathematicae 40.3 (2013): 309-326. <http://eudml.org/doc/286689>.

@article{MagdalenaMeller2013,
abstract = {We consider the smoothness parameter of a function f ∈ L²(ℝ) in terms of Besov spaces $B^\{s\}_\{2,∞\}(ℝ)$, $s*(f) = sup\{s > 0: f ∈ B^\{s\}_\{2,∞\}(ℝ)\}$. 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).},
author = {Magdalena Meller, Natalia Jarzębkowska},
journal = {Applicationes Mathematicae},
keywords = {estimation; Besov spaces; smoothness parameter; franklin- strömberg wavelet; spline wavelets},
language = {eng},
number = {3},
pages = {309-326},
title = {Estimation of a smoothness parameter by spline wavelets},
url = {http://eudml.org/doc/286689},
volume = {40},
year = {2013},
}

TY - JOUR
AU - Magdalena Meller
AU - Natalia Jarzębkowska
TI - Estimation of a smoothness parameter by spline wavelets
JO - Applicationes Mathematicae
PY - 2013
VL - 40
IS - 3
SP - 309
EP - 326
AB - We consider the smoothness parameter of a function f ∈ L²(ℝ) in terms of Besov spaces $B^{s}_{2,∞}(ℝ)$, $s*(f) = sup{s > 0: f ∈ B^{s}_{2,∞}(ℝ)}$. 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).
LA - eng
KW - estimation; Besov spaces; smoothness parameter; franklin- strömberg wavelet; spline wavelets
UR - http://eudml.org/doc/286689
ER -