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

Linear and nonlinear approximations to functions from Besov spaces B p, q σ ([0, 1]), σ > 0, 1 ≤ p, q ≤ ∞ 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(σ, σ + 1/2 - 1/p) for all 1 ≤ p ≤ ∞ and σ > 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 -σ for all 1 ≤ p ≤ ∞ and σ > max(1/p - 1/2, 0). The nonlinear approximation scheme proposed does not require any a priori information on the approximated function. © Nauka/Interperiodica 2006.