View source for Runge domain

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''of the first kind''

A domain $G$ in the space $\mathbf C^n$ of complex variables $(z_1,\dotsc,z_n)$ with the property that for any function $f(z_1,\dotsc,z_n)$ holomorphic in $G$ there exists a sequence of polynomials

$$\{P_k(z_1,\dotsc,z_n)\}_{k=1}^\infty\tag{1}$$

converging in $G$ to $f(z_1,\dotsc,z_n)$ uniformly on every closed bounded set $E\subset G$. The definition of a Runge domain of the second kind is obtained from the above by replacing the sequence \ref{1} by a sequence of rational functions $\{R_k\{z_1,\dotsc,z_n)\}_{k=1}^\infty$. For $n=1$ any simply-connected domain is a Runge domain of the first kind and any domain is a Runge domain of the second kind (see [[Runge theorem|Runge theorem]]). For $n\geq2$ not all simply-connected domains are Runge domains and not all Runge domains are simply connected.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.A. Fuks,   "Special chapters in the theory of analytic functions of several complex variables" , Amer. Math. Soc.  (1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.S. Vladimirov,   "Methods of the theory of functions of several complex variables" , M.I.T.  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G. Stolzenberg,   "Polynomially and rationally convex sets"  ''Acta Math.'' , '''109'''  (1963)  pp. 259–289</TD></TR></table>



====Comments====
Two domains $G$ and $D$ with $G\subset D$ are called a Runge pair if every function holomorphic in $G$ can be approximated uniformly on every compact subset of $G$ by functions holomorphic in $D$. One also says that $G$ is (relatively) Runge in $D$. To say that $G$ is a Runge domain (of the first kind) is equivalent to saying that $(G,\mathbf C^n)$ is a Runge pair.

In addition there are the following generalizations: If $G_1\subset G_2\subset\mathbf C^n$ are two domains, then $G_1$ is called relatively Runge in $G_2$ if every holomorphic function on $G_1$ can be uniformly approximated on every compact subset of $G_1$ with holomorphic functions on $G_2$. Hence $G$ is a Runge domain of the first kind if and only if $G$ is relatively Runge in $\mathbf C^n$.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.M. Range,   "Holomorphic functions and integral representation in several complex variables" , Springer  (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.E. Fornaess,   B. Stensønes,   "Lectures on counterexamples in several variables" , Princeton Univ. Press  (1987)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L. Hörmander,   "An introduction to complex analysis in several variables" , North-Holland  (1973)  pp. Sect. 4.5</TD></TR></table>