Runge domain

of the first kind

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

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

converging in $G$ to $f(z_1,\dots,z_n)$ 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 \eqref{1} by a sequence of rational functions $\{R_k\{z_1,\dots,z_n)\}_{k=1}^\infty$. 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). For $n\geq2$ not all simply-connected domains are Runge domains and not all Runge domains are simply connected.

References

[1]	B.A. Fuks, "Special chapters in the theory of analytic functions of several complex variables" , Amer. Math. Soc. (1965) (Translated from Russian)
[2]	V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)
[3]	G. Stolzenberg, "Polynomially and rationally convex sets" Acta Math. , 109 (1963) pp. 259–289

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

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