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$L^{p}$-APPROXIMATION OF HOLOMORPHIC FUNCTIONS ON A CLASS OF CONVEX DOMAINS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 97 / Issue 3 / June 2018
- Published online by Cambridge University Press:
- 23 April 2018, pp. 446-452
- Print publication:
- June 2018
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- Article
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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.
