Difference between revisions of "Runge theorem"
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A theorem on the possibility of polynomially approximating holomorphic functions, first proved by C. Runge (1885) (cf. also [[Approximation of functions of a complex variable|Approximation of functions of a complex variable]]). | A theorem on the possibility of polynomially approximating holomorphic functions, first proved by C. Runge (1885) (cf. also [[Approximation of functions of a complex variable|Approximation of functions of a complex variable]]). | ||
| − | Let | + | Let $D$ be a simply-connected domain in the complex $z$-plane. Then any function $f$ holomorphic in $D$ can be approximated uniformly on compact sets inside $D$ by polynomials in $z$. More precisely, for any compact set $K\subset D$ and $\epsilon>0$ there is a polynomial $p(z)$ with complex coefficients such that $|f(z)-p(z)|<\epsilon$ for all $z\in K$. |
| − | In other words: Any function | + | In other words: Any function $f$ holomorphic in a simply-connected domain $D\subset\mathbf C$ can be represented as a series of polynomials in $z$ converging absolutely and uniformly to $f$ on compact sets inside $D$. |
| − | An equivalent statement of Runge's theorem: Let | + | An equivalent statement of Runge's theorem: Let $K$ be a compact set in $\mathbf C$ with connected complement $\mathbf C\setminus K$; then any function holomorphic in a neighbourhood of $K$ can be approximated uniformly on $K$ by polynomials in $z$. In this formulation, Runge's theorem is a special case of Mergelyan's theorem (cf. [[Mergelyan theorem|Mergelyan theorem]]). |
| − | The following theorem on rational approximation is also called Runge's theorem: Any function | + | The following theorem on rational approximation is also called Runge's theorem: Any function $f$ holomorphic in a domain $D\subset\mathbf C$ can be uniformly approximated on compact sets inside $D$ by rational functions with poles outside $D$. |
Runge's theorem has many applications in the theory of functions of a complex variable and in functional analysis. A theorem analogous to Runge's theorem is valid for non-compact Riemann surfaces. An extension of Runge's theorem to functions of several complex variables is the Oka–Weil theorem (see [[Oka theorems|Oka theorems]]). | Runge's theorem has many applications in the theory of functions of a complex variable and in functional analysis. A theorem analogous to Runge's theorem is valid for non-compact Riemann surfaces. An extension of Runge's theorem to functions of several complex variables is the Oka–Weil theorem (see [[Oka theorems|Oka theorems]]). | ||
Latest revision as of 18:58, 1 May 2014
A theorem on the possibility of polynomially approximating holomorphic functions, first proved by C. Runge (1885) (cf. also Approximation of functions of a complex variable).
Let $D$ be a simply-connected domain in the complex $z$-plane. Then any function $f$ holomorphic in $D$ can be approximated uniformly on compact sets inside $D$ by polynomials in $z$. More precisely, for any compact set $K\subset D$ and $\epsilon>0$ there is a polynomial $p(z)$ with complex coefficients such that $|f(z)-p(z)|<\epsilon$ for all $z\in K$.
In other words: Any function $f$ holomorphic in a simply-connected domain $D\subset\mathbf C$ can be represented as a series of polynomials in $z$ converging absolutely and uniformly to $f$ on compact sets inside $D$.
An equivalent statement of Runge's theorem: Let $K$ be a compact set in $\mathbf C$ with connected complement $\mathbf C\setminus K$; then any function holomorphic in a neighbourhood of $K$ can be approximated uniformly on $K$ by polynomials in $z$. In this formulation, Runge's theorem is a special case of Mergelyan's theorem (cf. Mergelyan theorem).
The following theorem on rational approximation is also called Runge's theorem: Any function $f$ holomorphic in a domain $D\subset\mathbf C$ can be uniformly approximated on compact sets inside $D$ by rational functions with poles outside $D$.
Runge's theorem has many applications in the theory of functions of a complex variable and in functional analysis. A theorem analogous to Runge's theorem is valid for non-compact Riemann surfaces. An extension of Runge's theorem to functions of several complex variables is the Oka–Weil theorem (see Oka theorems).
References
| [1] | A.I. Markushevich, "A short course on the theory of analytic functions" , Moscow (1978) (In Russian) |
| [2] | B.V. Shabat, "Introduction of complex analysis" , 1 , Moscow (1976) (In Russian) |
Comments
For more about Runge's theorem and its generalizations, such as Walsh' theorem, the Keldysh theorem, the Lavrent'ev theorem, see [a6], [a1], Chapt. VIII, and [a3], Chapt. III, for the case of the complex plane; [a2], Sect. 25, for the case of Riemann surfaces; and [a4], Sect. 7, for the case of several complex variables.
References
| [a1] | R.B. Burchel, "An introduction to classical complex analysis" , 1 , Acad. Press (1979) |
| [a2] | O. Forster, "Lectures on Riemann surfaces" , Springer (1981) (Translated from German) |
| [a3] | D. Gaier, "Lectures on complex approximation" , Birkhäuser (1987) (Translated from German) |
| [a4] | J. Wermer, "Banach algebras and several complex variables" , Springer (1976) pp. Chapt. 13 |
| [a5] | W. Rudin, "Real and complex analysis" , McGraw-Hill (1978) pp. 24 |
| [a6] | A.I. Markushevich, "Theory of functions of a complex variable" , 3 , Chelsea (1977) pp. Sect. 3.12 (Translated from Russian) |
Runge theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Runge_theorem&oldid=13069