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

By Stéphanie Nivoche

Appears in collection : Complex Geometry, Dynamical Systems and Foliation Theory / Géométrie complexe, systèmes dynamiques et théorie de feuilletages

Given a domain D in $C^n$ and a compact subset K of D, the set $A^D_K$ of all restrictions of functions holomorphic on D the modulus of which is bounded by 1 is a compact subset of the Banach space $C(K)$ of continuous functions on K. The sequence $d_m(A^D_K)$ of Kolmogorov m-widths of $A^D_K$ provides a measure of the degree of compactness of the set $A^D_K$ in $C(K)$ and the study of its asymptotics has a long history, essentially going back to Kolmogorov's work on epsilon-entropy of compact sets in the 1950s. In the 1980s Zakharyuta gave, for suitable D and K, the exact asymptotics of these diameters (1), and showed that is implied by a conjecture, now known as Zakharyuta's Conjecture, concerning the approximability of the regularised relative extremal function of K and D 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 D strictly hyperconvex and K nonpluripolar 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 polyhedral.

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Citation data

  • DOI 10.24350/CIRM.V.19969703
  • Cite this video Nivoche Stéphanie (10/17/22). New solution of a problem of Kolmogorov on width asymptotics in holomorphic function spaces. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19969703
  • URL https://dx.doi.org/10.24350/CIRM.V.19969703

Bibliography

  • BANDTLOW, Oscar F. et NIVOCHE, Stéphanie. New solution of a problem of Kolmogorov on width asymptotics in holomorphic function spaces. Journal of the European Mathematical Society, 2021, vol. 24, no 7, p. 2493-2532. - https://doi.org/10.48550/arXiv.1906.00918

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