This is an announcement for the paper “Greedy Algorithms and Kolmogorov Widths in Banach Spaces” by Stephen J. Dilworthhttps://arxiv.org/find/math/1/au:+Dilworth_S/0/1/0/all/0/1, Van Kien Nguyenhttps://arxiv.org/search?searchtype=author&query=Nguyen%2C+V+K.
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 $V_n\subset X$ which can be used to approximate the elements of $K$. We are interested in how well the space $V_n$ approximates the elements of $K$. For this purpose we compare the greedy algorithm with the Kolmogorov, width which is the best possible error one can approximate $K$ by $n$−dimensional subspaces. Various results in this direction have been given, e.g., in Binev et al. (SIAM J. Math. Anal. (2011)), DeVore et al. (Constr. Approx. (2013)) and Wojtaszczyk (J. Math. Anal. Appl. (2015)). The purpose of the present paper is to continue this line. We shall show that under some additional assumptions the results in the above-mentioned papers can be improved.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1804.03935