Application of wavelet bases in linear and nonlinear approximation to functions from Besov spaces

Linear and nonlinear approximations to functions from Besov spaces $B^\sigma_{p,q}([0,1])$, $\sigma>0$, $1\le p,q\le\infty$, in a wavelet basis are considered. It is shown that an optimal linear approximation by a $D$-dimensional subspace of basis wavelet functions has an error of order $D^{-\min(\sigma,\sigma+1/2-1/p)}$ for all $1\le p\le\infty$ and $\sigma>\max(1/p-1/2,0)$. An original scheme is proposed for optimal nonlinear approximation. It is shown how a $D$-dimensional subspace of basis wavelet functions is to be chosen depending on the approximated function so that the error is on the order of $D^{-\sigma}$ for all $1\le p\le\infty$ and $\sigma>\max(1/p-1/2,0)$ . The nonlinear approximation scheme proposed does not require any a priori information on the approximated function.