Geodesic
Uniform Approximation of Functions Continuous on a Compact Subset of $\mathbb C$ and Analytic in Its Interior by Functions Bianalytic in Its Neighborhoods
Matematičeskie zametki, Tome 69 (2001) no. 2, pp. 245-261

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We prove that an arbitrary function continuous on a compact set $X\subset\mathbb C$ and holomorphic in the interior of $X$ can be approximated by functions bianalytic in neighborhoods of $X$ with arbitrary accuracy. 
@article{MZM_2001_69_2_a8,
     author = {M. Ya. Mazalov},
     title = {Uniform {Approximation} of {Functions} {Continuous} on a {Compact} {Subset} of $\mathbb C$ and {Analytic} in {Its} {Interior} by {Functions} {Bianalytic} in {Its} {Neighborhoods}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {245--261},
     publisher = {mathdoc},
     volume = {69},
     number = {2},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2001_69_2_a8/}
}
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M. Ya. Mazalov. Uniform Approximation of Functions Continuous on a Compact Subset of $\mathbb C$ and Analytic in Its Interior by Functions Bianalytic in Its Neighborhoods. Matematičeskie zametki, Tome 69 (2001) no. 2, pp. 245-261. http://geodesic.mathdoc.fr/item/MZM_2001_69_2_a8/