On greedy algorithm approximating Kolmogorov widths in Banach spaces

The greedy algorithm to produce n-dimensional subspaces X-n to approximate a compact set F contained in a Hilbert space was introduced in the context of reduced basis method in [12,13]. The same algorithm works for a general Banach space and in this context was studied in [4]. In this paper we study the case F subset of L-p. If Kolmogorov diameters d(n)(F) of F decay as n(-alpha) we give an almost optimal estimate for the decay of sigma(n) := dist(F,X-n). We also give some direct estimates of the form sigma(n) <= C(n)d(n)(F). (C) 2014 Published by Elsevier Inc.