Multivariate n-term rational and piecewise polynomial approximation

We study nonlinear approximation in Lp(R d) (0 < p < ∞, d> 1) from (a) n-term rational functions, and (b) piecewise polynomials generated by different anisotropic dyadic partitions of R d. To characterize the rates of each such piecewise polynomial approximation we introduce a family of smoothness spaces (B-spaces) which can be viewed as an anisotropic variation of Besov spaces. We use the B-spaces to prove Jackson and Bernstein estimates and then characterize the piecewise polynomial approximation by interpolation. Our main estimate relates n-term rational approximation with piecewise polynomial approximation in Lp(R d). This result enables us to obtain a direct estimate for n-term rational approximation in terms of a minimal B-norm (over all dyadic partitions). We also show that the Haar bases associated with anisotropic dyadic partitions of R d can be successfully utilized for nonlinear approximation. We give an effective algorithm for best Haar basis or best B-space selection.