Weierstrasse theorem of convergence

The Weierstrass convergence theorem is a by Karl Weierstrass named set from the function theory . It says that the limit function of a locally uniformly convergent sequence of holomorphic functions is in turn a holomorphic function. In addition, all derivatives also converge locally evenly against the corresponding derivative of the limit function.

formulation

Let be a domain and a sequence of holomorphic functions that converges uniformly to a function on locally , that is, for each there is a neighborhood of such that on converges uniformly on to . Then: ${\displaystyle G\subset \mathbb {C} }$ ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle f_{n}\colon G\rightarrow \mathbb {C} }$ ${\displaystyle G}$ ${\displaystyle f\colon G\rightarrow \mathbb {C} }$ ${\displaystyle z\in G}$ ${\displaystyle U\subset G}$ ${\displaystyle z}$ ${\displaystyle f_{n}}$ ${\displaystyle U}$ ${\displaystyle f}$

• ${\displaystyle f}$ is holomorphic.

• For each converges evenly to locally on . ${\displaystyle k\in \mathbb {N} _{0}}$ ${\displaystyle (f_{n}^{(k)})_{n\in \mathbb {N} }}$ ${\displaystyle G}$ ${\displaystyle f^{(k)}}$

Counterexamples in the real

Weierstraß's theorem of convergence is remarkable in that its real analog is false: the limit function of a uniformly convergent sequence of differentiable functions does not have to be differentiable, and even if it is, the derivatives of the sequence members need not converge point by point to the derivative of the limit function.

• Fix a continuous but nowhere differentiable function . According to Weierstrass's approximation theorem, there is a sequence of polynomials which converges uniformly on against . ${\displaystyle f\colon [0,1]\rightarrow \mathbb {R} }$ ${\displaystyle [0,1]}$ ${\displaystyle f}$

• The sequence converges uniformly to the null function , while the derivatives nowhere converge to the derivative of the null function. ${\displaystyle f_{n}(x):={\frac {1}{n}}\sin(nx)}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle f_{n}'(x)=\cos(nx)}$

literature

• Eberhard Freitag, Rolf Busam: Function Theory 1 . 3. Edition. Springer-Verlag 2000, ISBN 3540676414 .