Runge,s Theorem
المؤلف:
Szegö, G
المصدر:
Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc
الجزء والصفحة:
...
17-11-2021
1590
Runge's Theorem
Let 
be compact, let 
be analytic on a neighborhood of 
, and let 
contain at least one point from each connected component of 
. Then for any 
, there is a rational function 
with poles in 
such that
(Krantz 1999, p. 143).
A polynomial version can be obtained by taking ![P=<span style=]()
{infty}" src="https://mathworld.wolfram.com/images/equations/RungesTheorem/Inline9.gif" style="height:15px; width:48px" />. Let 
be an analytic function which is regular in the interior of a Jordan curve 
and continuous in the closed domain bounded by 
. Then 
can be approximated with arbitrary accuracy by polynomials (Szegö 1975, p. 5; Krantz 1999, p. 144).
REFERENCES:
Krantz, S. G. "Runge's Theorem." §11.1.2 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 143-144, 1999.
Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., p. 7, 1975.
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