Runge domain

of the first kind

A domain in the space of complex variables with the property that for any function holomorphic in there exists a sequence of polynomials

(1)

converging in to uniformly on every closed bounded set . The definition of a Runge domain of the second kind is obtained from the above by replacing the sequence (1) by a sequence of rational functions . For 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 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 and with are called a Runge pair if every function holomorphic in can be approximated uniformly on every compact subset of by functions holomorphic in . One also says that is (relatively) Runge in . To say that is a Runge domain (of the first kind) is equivalent to saying that is a Runge pair.

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

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