Abstract
In discs B, all holomorphic functions are approximated compactly by their Taylor polynomials. In particular, for every f ∈ O(B) and every compact set K in B, there exists a sequence of polynomials pnsuch that lim |f - Pn|K = 0. In arbitrary domains, polynomial approximation is not always possible; in C×, for example, there is no sequence of polynomials pn that approximates the holomorphic function l/z uniformly on a circle γ, for it would then follow that
Runge approximation theory charms by its wonderful balance between freedom and necessity.
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Remmert, R. (1998). Runge Theory for Compact Sets. In: Classical Topics in Complex Function Theory. Graduate Texts in Mathematics, vol 172. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-2956-6_12
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DOI: https://doi.org/10.1007/978-1-4757-2956-6_12
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