Relative widths of Sobolev classes in the uniform and integral metrics

Abstract

Let W p r be the Sobolev class consisting of 2π-periodic functions f such that ‖f (r)‖ p ≤ 1. We consider the relative widths d n (W p r , MW p r , L p ), which characterize the best approximation of the class W p r in the space L p by linear subspaces for which (in contrast to Kolmogorov widths) it is additionally required that the approximating functions g should lie in MW p r , i.e., ‖g (r)‖ p ≤ M. We establish estimates for the relative widths in the cases of p = 1 and p = ∞; it follows from these estimates that for almost optimal (with error at most Cn −r , where C is an absolute constant) approximations of the class W p r by linear 2n-dimensional spaces, the norms of the rth derivatives of some approximating functions are not less than cln min(n, r) for large n and r.