On greedy algorithm approximating Kolmogorov widths in Banach spaces

The greedy algorithm to produce n  -dimensional subspaces XnXn to approximate a compact set FF contained in a Hilbert space was introduced in the context of reduced basis method in [12] and [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⊂LpF⊂Lp. If Kolmogorov diameters dn(F)dn(F) of FF decay as n−αn−α we give an almost optimal estimate for the decay of σn:=dist(F,Xn)σn:=dist(F,Xn). We also give some direct estimates of the form σn≤Cndn(F)σn≤Cndn(F).