Comparison of wavelet approximation order in different smoothness spaces

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.