On Exact Values of n-Widths in a Hilbert Space

Abstract

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 functions, and, on the other hand, classes of functions in a Hilbert space which are elliptical cylinders or generalized octahedra. This second case is surveyed and new results are presented. For n -widths of ellipsoids, elliptic cylinders, and generalized octahedra, upper bounds for the n -widths are based on the Fourier method. The lower bounds are based on the method of “embedded balls” for ellipsoids and the method of averaging for generalized octahedra. General theorems concerning elliptical cylinders and generalized octahedra are proved, various corollaries from these general theorems are considered, and some additional problems (average n -widths, extremal spaces for an ellipsoids and octahedra, etc.) are discussed.