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''of the first kind''
 
''of the first kind''
  
A domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082800/r0828001.png" /> in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082800/r0828002.png" /> of complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082800/r0828003.png" /> with the property that for any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082800/r0828004.png" /> holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082800/r0828005.png" /> there exists a sequence of polynomials
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082800/r0828006.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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$$\{P_k(z_1,\dots,z_n)\}_{k=1}^\infty\label{1}\tag{1}$$
  
converging in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082800/r0828007.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082800/r0828008.png" /> uniformly on every closed bounded set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082800/r0828009.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082800/r08280010.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082800/r08280011.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082800/r08280012.png" /> not all simply-connected domains are Runge domains and not all Runge domains are simply connected.
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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|Runge theorem]]). For $n\geq2$ not all simply-connected domains are Runge domains and not all Runge domains are simply connected.
  
 
====References====
 
====References====
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====Comments====
 
====Comments====
Two domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082800/r08280013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082800/r08280014.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082800/r08280015.png" /> are called a Runge pair if every function holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082800/r08280016.png" /> can be approximated uniformly on every compact subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082800/r08280017.png" /> by functions holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082800/r08280018.png" />. One also says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082800/r08280019.png" /> is (relatively) Runge in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082800/r08280020.png" />. To say that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082800/r08280021.png" /> is a Runge domain (of the first kind) is equivalent to saying that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082800/r08280022.png" /> is a Runge pair.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082800/r08280023.png" /> are two domains, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082800/r08280024.png" /> is called relatively Runge in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082800/r08280025.png" /> if every holomorphic function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082800/r08280026.png" /> can be uniformly approximated on every compact subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082800/r08280027.png" /> with holomorphic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082800/r08280028.png" />. Hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082800/r08280029.png" /> is a Runge domain of the first kind if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082800/r08280030.png" /> is relatively Runge in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082800/r08280031.png" />.
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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====
 
====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>
 
<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>

Latest revision as of 15:09, 14 February 2020

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
How to Cite This Entry:
Runge domain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Runge_domain&oldid=12515
This article was adapted from an original article by E.P. Dolzhenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article