Relative widths of Sobolev classes in the uniform and integral metrics

Let W (p) (r) be the Sobolev class consisting of 2 pi-periodic functions f such that aEuro-f ((r))aEuro- (p) a parts per thousand currency sign 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., aEuro-g ((r))aEuro- (p) a parts per thousand currency sign M. We establish estimates for the relative widths in the cases of p = 1 and p = a; 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.