JournalsjemsVol. 24, No. 7pp. 2493–2532

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

  • Oscar F. Bandtlow

    Queen Mary University of London, UK
  • Stéphanie Nivoche

    Université Côte d’Azur, Nice, France
New solution of a problem of Kolmogorov on width asymptotics in holomorphic function spaces cover
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Abstract

Given a domain DD in Cn\mathbb{C}^n and a compact subset KK of DD, the set AKD\mathcal{A}_K^D of all restrictions of functions holomorphic on DD the modulus of which is bounded by 11 is a compact subset of the Banach space C(K)C(K) of continuous functions on KK. The sequence (dm(AKD))mN(d_m(\mathcal{A}_K^D))_{m\in \mathbb{N}} of Kolmogorov mm-widths of AKD\mathcal{A}_K^D provides a measure of the degree of compactness of the set AKD\mathcal{A}_K^D in C(K)C(K) and the study of its asymptotics has a long history, essentially going back to Kolmogorov’s work on ϵ\epsilon-entropy of compact sets in the 1950s. In the 1980s Zakharyuta showed that for suitable DD and KK the asymptotics

limmlogdm(AKD)m1/n=2π(n!C(K,D))1/n,\lim_{m\to \infty}\frac{- \log d_m(\mathcal{A}_K^D)}{m^{1/n}} = 2\pi \bigg( \frac{n!}{C(K,D)} \bigg) ^{1/n},

where C(K,D)C(K,D) is the Bedford–Taylor relative capacity of KK in DD, is implied by a conjecture, now known as Zakharyuta's Conjecture, concerning the approximability of the regularised relative extremal function of KK and DD by certain pluricomplex Green functions. Zakharyuta’s Conjecture was proved by Nivoche in 2004 thus settling (1) at the same time.

We shall give a new proof of the asymptotics (1) for DD strictly hyperconvex and KK non-pluripolar which does not rely on Zakharyuta’s Conjecture. Instead we proceed more directly by a two-pronged approach establishing sharp upper and lower bounds for the Kolmogorov widths. The lower bounds follow from concentration results of independent interest for the eigenvalues of a certain family of Toeplitz operators, while the upper bounds follow from an application of the Bergman–Weil formula together with an exhaustion procedure by special holomorphic polyhedra.

Cite this article

Oscar F. Bandtlow, Stéphanie Nivoche, New solution of a problem of Kolmogorov on width asymptotics in holomorphic function spaces. J. Eur. Math. Soc. 24 (2022), no. 7, pp. 2493–2532

DOI 10.4171/JEMS/1148