Blaschke's convergence theorem

The convergence rate of Blaschke is a theorem of complex analysis , which on the Austrian mathematician Wilhelm Blaschke back. The theorem is closely related to Vitali's convergence theorem . It gives a criterion for the convergence of sequences of holomorphic functions on the open unit disk .

In his original evidence from 1915, Wilhelm Blaschke relies heavily on Montel’s selection principle . In 1923, Tibor Radó and Karl Löwner provided a direct proof of the theorem using explicit estimates .

Formulation of the convergence theorem

The sentence can be formulated as follows:

Given a sequence of holomorphic functions on the unit disk , which is uniformly bounded : ${\displaystyle \mathbb {E} =\{z\colon \,|z|<1\}\subset \mathbb {C} }$ ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle {f_{n}}\colon \,\mathbb {E} \to \mathbb {C} }$ ${\displaystyle (n\in \mathbb {N} )}$

${\displaystyle \sup _{n\in \mathbb {N} ,z\in \mathbb {E} }{|{f_{n}(z)}|}<\infty }$.

For this purpose, there is a sequence of different numbers such that the following conditions are met: ${\displaystyle z_{k}\in \mathbb {E} }$ ${\displaystyle (k\in \mathbb {N} )}$

1. ${\displaystyle \sum _{k=1}^{\infty }(1-{|z_{k}|})=\infty }$.
2. For each   exists . ${\displaystyle k\in \mathbb {N} }$     ${\displaystyle \lim _{n\rightarrow \infty }f_{n}(z_{k})}$

Then the sequence of functions converges in compact . ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \mathbb {E} }$

literature

Original work

• W. Blaschke: An extension of Vitali's theorem about sequences of analytical functions . In: Ber. Negotiating Kings Saxon. Ges. Wiss. Leipzig . 67, 1915, pp. 194-200.
• K. Lowner; T. Radó: Comment on Blaschke's Convergence Theorems . In: Jahresber. German. Math. Association . 32, 1923, pp. 198-200.

Monographs

• Robert B. Burckel: An Introduction to Classical Complex Analysis . Birkhäuser Verlag, Basel [u. a.] 1979, ISBN 3-7643-0989-X .
• Reinhold Remmert , Georg Schumacher: Function theory 2 (=  Springer textbook - basic knowledge of mathematics ). 3rd, revised edition. Springer-Verlag, Berlin [a. a.] 2007, ISBN 978-3-540-40432-3 .

Individual evidence

1. ↑ Blaschke: An extension of Vitali's theorem on sequences of analytical functions . In: Ber. Negotiating Kings Saxon. Ges. Wiss. Leipzig . tape 67 , 1915, pp. 194 ff .
2. ↑ Burckel: pp. 219, 469.
3. ↑ ^{a b} Remmert, Schumacher: p. 151.
4. ↑ Radó, Löwner: Comment on a Blaschke's convergence theorems. In: Jahresber. DMV . tape 32 , 1923, pp. 198 ff .
5. ↑ ^{a b} Burckel: p. 219.