Abstract
We prove that a holomorphic function on a neighborhood of a compact convex set \(K \subset {{\,\mathrm{{\mathbb {C}}}\,}}^n\) can be uniformly on K approximated by polynomials with an error that decreases exponentially fast with the growth of the polynomial degree. The presented method is based on the vanishing of the top Dolbeault cohomology group of an open subset in \({{\,\mathrm{{\mathbb {C}}}\,}}^n\) and an argument involving Čech cohomology. In comparison with the Bernstein-Walsh approach previously applied to the problems of this type the method presented here is much more elementary but it does not provide effective estimates.
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Acknowledgements
The author was supported by the Moscow Center of Fundamental and Applied Mathematics at INM RAS (Agreement with the Ministry of Education and Science of the Russian Federation No. 075-15-2022-286). The author also expresses gratitude to his colleagues N. Zamarashkin and A. Zyl, who introduced him to the problems of rapid polynomial approximation and tensor ranks estimations.
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Matvey Smirnov wrote the manuscript text and reviewed the manuscript.
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Communicated by Irene Sabadini.
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Smirnov, M. On the Rate of Polynomial Approximations of Holomorphic Functions on Convex Compact Sets. Complex Anal. Oper. Theory 17, 129 (2023). https://doi.org/10.1007/s11785-023-01430-z
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DOI: https://doi.org/10.1007/s11785-023-01430-z