Greedy algorithms and Kolmogorov widths in Banach spaces

Abstract

Let X be a Banach space and K be a compact subset in X. We consider a greedy algorithm for finding n-dimensional subspace Vn⊂X which can be used to approximate the elements of K. We are interested in how well the space Vn approximates the elements of K. For this purpose we compare the performance of greedy algorithm measured by σn(K)X≔dist(K,Vn)X with the Kolmogorov width dn(K)X which is the best possible error one can achieve when approximating K by n-dimensional subspaces. Various results in this direction have been given, e.g., in Binev et al. (2011), DeVore et al. (2013) and Wojtaszczyk (2015). The purpose of the present paper is to continue this line. We shall show that there exists a constant C>0 such that σn(K)X≤Cn−s+μ(log(n+1))min(s,1∕2),n≥1, if Kolmogorov widths dn(K)X decay as n−s and the Banach–Mazur distance between an arbitrary n-dimensional subspace Vn⊂X and ℓ2n satisfies d(Vn,ℓ2n)≤C1nμ. In particular, when some additional information about the set K is given then there is no logarithmic factor in this estimate.