On the Rate of Polynomial Approximations of Holomorphic Functions on Convex Compact Sets

We prove that a holomorphic function on a neighborhood of a compact convex set K ⊂ 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 cohom...

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Bibliographic Details
Published in:Complex analysis and operator theory Vol. 17; no. 8; p. 129
Main Author: Smirnov, Matvey
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 01.11.2023
Springer Nature B.V
Subjects:
Analysis
Analytic functions
Approximation
Convexity
Higher Dimensional Geometric Function Theory and Hypercomplex Analysis
Homology
Mathematics
Mathematics and Statistics
Neighborhoods
Numerical analysis
Operator Theory
Polynomials
Several complex variables
Čech cohomology
32A08
Polynomial approximation
32A10
ISSN:1661-8254, 1661-8262
Online Access:Get full text
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Summary:We prove that a holomorphic function on a neighborhood of a compact convex set K ⊂ 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 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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ISSN:1661-8254
1661-8262
DOI:10.1007/s11785-023-01430-z