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Search: All articles in the CMB digital archive with keyword Carleman

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1. CMB Online first

Gauthier, Paul M; Sharifi, Fatemeh
Luzin-type holomorphic approximation on closed subsets of open Riemann surfaces
It is known that if $E$ is a closed subset of an open Riemann surface $R$ and $f$ is a holomorphic function on a neighbourhood of $E,$ then it is ``usually" not possible to approximate $f$ uniformly by functions holomorphic on all of $R.$ We show, however, that for every open Riemann surface $R$ and every closed subset $E\subset R,$ there is closed subset $F\subset E,$ which approximates $E$ extremely well, such that every function holomorphic on $F$ can be approximated much better than uniformly by functions holomorphic on $R$.

Keywords:Carleman approximation, tangential approximation, Myrberg surface
Categories:30E15, 30F99

2. CMB 2015 (vol 59 pp. 87)

Gauthier, Paul M.; Kienzle, Julie
Approximation of a Function and its Derivatives by Entire Functions
A simple proof is given for the fact that, for $m$ a non-negative integer, a function $f\in C^{(m)}(\mathbb{R}),$ and an arbitrary positive continuous function $\epsilon,$ there is an entire function $g,$ such that $|g^{(i)}(x)-f^{(i)}(x)|\lt \epsilon(x),$ for all $x\in\mathbb{R}$ and for each $i=0,1\dots,m.$ We also consider the situation, where $\mathbb{R}$ is replaced by an open interval.

Keywords:Carleman theorem
Category:30E10

3. CMB 2002 (vol 45 pp. 80)

Gauthier, P. M.; Zeron, E. S.
Approximation On Arcs and Dendrites Going to Infinity in $\C^n$
On a locally rectifiable arc going to infinity, each continuous function can be approximated by entire functions.

Keywords:tangential approximation, Carleman
Categories:32E30, 32E25

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