International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 63670, 7 pages
doi:10.1155/IJMMS/2006/63670
Comparison of wavelet approximation order in different smoothness spaces
Mathematics Discipline, Khulna University, Khulna 9208, Bangladesh
Received 20 April 2005; Revised 3 April 2006; Accepted 25 April 2006
Copyright © 2006 M. R. Islam et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
In linear approximation by wavelet, we approximate a given
function by a finite term from the wavelet series. The
approximation order is improved if the order of smoothness of the
given function is improved, discussed by Cohen
(2003), DeVore (1998), and Siddiqi (2004). But in the case of
nonlinear approximation, the approximation order is improved
quicker than that in linear case. In this
study we proved this assumption only for the Haar wavelet. Haar
function is an example of wavelet and this fundamental example
gives major feature of the general wavelet. A nonlinear space
comes from arbitrary selection of wavelet coefficients, which
represent the target function almost equally. In this case our
computational work will be reduced tremendously in the sense that
approximation error decays more quickly than that in
linear case.