English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Given a domain $D$ in $\mathbb{C}^n$ and $K$ a compact subset of $D$, the set$\mathcal{A}_K^D$ of all restrictions of functions holomorphic on $D$ themodulus of which is bounded by $1$ is a compact subset of the Banach space$C(K)$ of continuous functions on $K$. The sequence$(d_m(\mathcal{A}_K^D))_{m\in \mathbb{N}}$ of Kolmogorov $m$-widths of$\mathcal{A}_K^D$ provides a measure of the degree of compactness of the set$\mathcal{A}_K^D$ in $C(K)$ and the study of its asymptotics has a longhistory, essentially going back to Kolmogorov's work on $\epsilon$-entropy ofcompact sets in the 1950s. In the 1980s Zakharyuta showed that for suitable $D$and $K$ the asymptotics \begin{equation*}  \lim_{m\to \infty}\frac{- \log d_m(\mathcal{A}_K^D)}{m^{1/n}} =  2\pi \left ( \frac{n!}{C(K,D)}\right ) ^{1/n}\,, \end{equation*} where$C(K,D)$ is the Bedford-Taylor relative capacity of $K$ in $D$ is implied by aconjecture, now known as Zakharyuta's Conjecture, concerning theapproximability of the regularised relative extremal function of $K$ and $D$ bycertain pluricomplex Green functions. Zakharyuta's Conjecture was proved byNivoche in 2004 thus settling the asymptotics above at the same time.  We shall give a new proof of the asymptotics above for $D$ strictlyhyperconvex and $K$ non-pluripolar which does not rely on Zakharyuta'sConjecture. Instead we proceed more directly by a two-pronged approachestablishing sharp upper and lower bounds for the Kolmogorov widths. The lowerbounds follow from concentration results of independent interest for theeigenvalues of a certain family of Toeplitz operators, while the upper boundsfollow from an application of the Bergman-Weil formula together with anexhaustion procedure by special holomorphic polyhedra.

New solution of a problem of Kolmogorov on width asymptotics in holomorphic function spaces