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 |
Runge domain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Runge_domain&oldid=33001