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March 2021 A note on simultaneous approximation on Vitushkin sets
Raymond Mortini, Rudolf Rupp
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Hiroshima Math. J. 51(1): 57-63 (March 2021). DOI: 10.32917/h2020009

Abstract

Given a planar Jordan domain $G$ with rectifiable boundary, it is well known that smooth functions on the closure of $G$ do not always admit smooth extensions to $\mathbb C$. Further conditions on the boundary are necessary to guarantee such extensions. On the other hand, Weierstrass’ approximation theorem yields polynomials converging uniformly to $f \in C(\overline{G}, \mathbb C)$. In this note we show that for Vitushkin sets $K$ with $K = \overline{K^\circ}$ it is always possible to uniformly approximate on $K$ the smooth function $f \in C^1(K, \mathbb C)$ by smooth functions $f_n$ in $\mathbb C$ so that also $\overline{\partial}f_n$ converges uniformly to $\overline{\partial}f$ on $K$. As a byproduct we deduce from its ‘‘smooth in a neighborhood version’’ the general Gauss integral theorem for functions whose partial derivatives in $G$ merely admit continuous extensions to its boundary.

Acknowledgements

We thank Peter Pflug (Oldenburg, Germany) for providing reference [5] and Jochen Wengenroth (Trier, Germany) for useful comments and reference [8] in connection with smooth extensions of smooth functions.

Citation

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Raymond Mortini. Rudolf Rupp. "A note on simultaneous approximation on Vitushkin sets." Hiroshima Math. J. 51 (1) 57 - 63, March 2021. https://doi.org/10.32917/h2020009

Information

Received: 5 February 2020; Revised: 13 May 2020; Published: March 2021
First available in Project Euclid: 19 April 2021

Digital Object Identifier: 10.32917/h2020009

Subjects:
Primary: 30E10
Secondary: 26B20

Rights: Copyright © 2021 Hiroshima University, Mathematics Program

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Vol.51 • No. 1 • 2021
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