Estimation of a smoothness parameter by spline wavelets

Magdalena Meller¹ ; Natalia Jarzębkowska¹

¹ Faculty of Applied Mathematics Technical University of Gdańsk G. Narutowicza 11/12 80-952 Gdańsk, Poland

Applicationes Mathematicae, Tome 40 (2013) no. 3, pp. 309-326

Résumé

We consider the smoothness parameter of a function $f\in L^2(\mathbb {R})$ in terms of Besov spaces $B^s_{2,\infty }(\mathbb {R})$, \[ s^*(f)=\sup\{s>0: f\in B^s_{2,\infty }(\mathbb {R})\}. \] 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 ($p\geq 1$) 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 $(p\geq 1).$

Détail

DOI : 10.4064/am40-3-4

Keywords: consider smoothness parameter function mathbb terms besov spaces infty mathbb * sup infty mathbb existing results estimation smoothness dziedziul kucharska wolnik nbsp nonparametric statist employ haar basis limited * using p regular geq spline wavelets exponential decay extend density functions * applying franklin str mberg wavelet prove presented estimator * consistent piecewise constant functions furthermore results franklin str mberg wavelet generalised spline wavelet geq