Kolmogorov widths under holomorphic mappings (Q2794696)

The Kolmogorov \(n\)-width of a set \(K\) in a normed linear space \(X\) is defined by: NEWLINE\[NEWLINEd_n(K)_X = \inf_{\text{dim}\, W =n} \sup_{x\in K} \inf_{w\in W} \|x-w\|,NEWLINE\]NEWLINE where \(W\) varies over all \(n\)-dimensional linear subspaces of \(X\). If \(L\) is a bounded linear operator from the space \(X\) into the space \(Y\), and \(K\) is a compact subset of \(X\), then it is easily verified that NEWLINE\[NEWLINEd_n(L(K))_Y \leq \|L\| d_n(K)_X.NEWLINE\]NEWLINE A mapping \(u\) from \(X\) to \(Y\) is said to be holomorphic on an open set \(O\subset X\) if for each \(x\in O\), \(u\) has a Fréchet derivative at \(x\).NEWLINENEWLINEIn this paper the authors provide a partial extension of the above inequality in that they prove that if \(u\) is a holomorphic mapping from an open set \(O \subset X\) into \(Y\), \(u\) is uniformly bounded on \(O\), and \(K\subset O\) is a compact subset of \(X\), then for any \(s>1\) and \(t<s-1\) NEWLINE\[NEWLINE\sup_{n \geq 1} n^s d_n(K)_X <\infty \Rightarrow \sup_{n \geq 1} n^t d_n(u(K))_Y <\infty.NEWLINE\]NEWLINE This result is useful because in many instances the \(n\)-widths of \(K\) can be easily estimated, while those of \(u(K)\) can not. The motivation for this result are in various theorems concerning Kolmogorov \(n\)-widths of manifolds of solutions of certain parametrized PDEs.