An extrapolation theorem for nonlinear approximation and its applications (Q863340)

Let \(\{\phi_j\}_{j=0}^\infty\) be a sequence of \(L_\infty\) functions on a probability space \(\Omega\). For a given integer \(n\geq 0\), let \(\Pi_n\) denote the linear span of \(\{\phi_j\}_{j=0}^n\) and let \(B_p^n\) be the unit ball of \(\Pi_n\) in \(L_p(\Omega)\). Let further \(\sigma_n(f)_p\) be the error of the best possible \(L_p(\Omega)\) approximation to \(f\) by a linear combination of at most \(n\) functions \(\phi_j\), not necessarily the first \(n\). For a class \(W\) of functions \(f\) let \(\sigma_n(W)_p=\sup_{f \in W} \sigma_n(f)_p\). Under some mild conditions concerning the system \(\{\phi_j\}\), the author finds relations between various quantities of the form \(\sigma(B_r^n)_p\). For example, if \(1\leq r_1 \leq r_2 \leq p \leq \infty\), \(1 \leq m \leq n\), then \[ \sigma_m (B_{r_2}^n)_p \leq C \left ( \sigma_m (B_{r_1}^{2n})_p \right )^\gamma, \qquad \gamma=(1/2-1/p)(1/r_1-1/p)^{-1}. \] As applications, estimates are given for the \(m\)-widths of the Sobolev classes on the unit sphere of \({\mathbb R}^d\) and for approximations of weighted Besov classes by ultraspherical polynomials.