Multivariate n-term rational and piecewise polynomial approximation

We study nonlinear approximation in Lp(R) (0 < p < ∞, d > 1) from (a) n-term rational functions, and (b) piecewise polynomials generated by different anisotropic dyadic partitions of Rd. 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). 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 Rd can be successfully utilized for nonlinear approximation. We give an effective algorithm for best Haar basis or best B-space selection.