On exact values of n-widths in a Hilbert space

The exact values of Kolmogorov n-widths have been calculated for two basic classes of functions. They are, on the one hand, classes of real functions defined by variation diminishing kernels and similar classes of analytic fu nctions, and, on the other hand, classes of functions in a Hilbert space wh ich are elliptical cylinders or generalized octahedra. This second case is surveyed and new results are presented. For n-widths of ellipsoids, ellipti c cylinders, and generalized octahedra, upper bounds for the n-widths are b ased on the Fourier method. The lower bounds are based on the method of "em bedded balls" for ellipsoids and the method of averaging for generalized oc tahedra. General theorems concerning elliptical cylinders and generalized o ctahedra are proved, various corollaries from these general theorems are co nsidered, and some additional problems (average n-widths, external spaces f or an ellipsoids and octahedra, etc.) are discussed. (C) 2001 Academic Pres s.