On the interconnection of local and global approximations by holomorphic functions

It is proved that if a function $f\in\operatorname{Lip}(\alpha,X)$, $\alpha>2/3$, can be approximated locally outside its zero set by holomorphic functions, then it can be approximated also on the whole compact set $X$. This implies that if $f\in\operatorname{Lip}(\alpha,X)$, $\alpha>2/3$, and $f^2$ can be approximated by holomorphic functions on $X$, then so can $f$.
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