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$L^{p}$-APPROXIMATION OF HOLOMORPHIC FUNCTIONS ON A CLASS OF CONVEX DOMAINS

Part of: Holomorphic convexity Pseudoconvex domains Approximations and expansions Differential operators in several variables

Published online by Cambridge University Press:  23 April 2018

LY KIM HA Open the ORCID record for LY KIM HA [Opens in a new window]
LY KIM HA*
Affiliation:
Faculty of Mathematics and Computer Science, University of Science, Vietnam National University, HoChiMinh City (VNU-HCM), 227 Nguyen Van Cu street, District 5, Ho Chi Minh City, Vietnam email lkha@hcmus.edu.vn
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Abstract

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Let $\unicode[STIX]{x1D6FA}$ be a member of a certain class of convex ellipsoids of finite/infinite type in $\mathbb{C}^{2}$. In this paper, we prove that every holomorphic function in $L^{p}(\unicode[STIX]{x1D6FA})$ can be approximated by holomorphic functions on $\bar{\unicode[STIX]{x1D6FA}}$ in $L^{p}(\unicode[STIX]{x1D6FA})$-norm, for $1\leq p<\infty$. For the case $p=\infty$, the continuity up to the boundary is additionally required. The proof is based on $L^{p}$ bounds in the additive Cousin problem.

Keywords

Lp approximationholomorphic functionsHenkin solutionsthe additive Cousin problemfinite/infinite type

MSC classification

Primary: 32E30: Holomorphic and polynomial approximation, Runge pairs, interpolation
Secondary: 32T25: Finite type domains 32W05: $overlinepartial$ and $overlinepartial$-Neumann operators 41A63: Multidimensional problems (should also be assigned at least one other classification number in this section) 41A65: Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 97 , Issue 3 , June 2018 , pp. 446 - 452
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

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