Hadamard's series of gaps

Hadamard's gap series is a term from the mathematical branch of function theory . This term refers to a complex-valued power series with development point zero , the coefficients a gap condition sufficient (see below)

The term refers to the French mathematician Jacques Hadamard (1865–1963), who in an important work from 1892 examined the relationships between the singularities of a holomorphic function and the coefficients of its Taylor expansion . The general problem underlying here is the question of the boundary behavior of power series . It is about the question of the holomorphic areas of the holomorphic functions belonging to a given power series and the extent to which the edge of the convergence circle of a power series represents its natural limit .

Gap condition

A (Hadamard's) gap series is a complex power series that fulfills the following property: ${\displaystyle P(z)=\sum _{n=0}^{\infty }{a_{n}}z^{n}}$

(HL) There is a strictly monotonically increasing number sequence     of natural numbers and a real number   such that: ${\displaystyle (m_{n})_{n=0,1,2,\dots }}$   ${\displaystyle \delta >0}$

(HL1) For     is always    . ${\displaystyle n=0,1,2,\dots }$ ${\displaystyle m_{n+1}-m_{n}>{\delta }\cdot m_{n}}$

(HL2) For     are always     and at the same time    , if    . ${\displaystyle k,n=0,1,2,\dots }$ ${\displaystyle a_{m_{n}}\neq 0}$ ${\displaystyle a_{k}=0}$ ${\displaystyle m_{n}<k<m_{n+1}}$

The following applies:

(HL *)     with   ${\displaystyle P(z)=\sum _{n=0}^{\infty }a_{m_{n}}z^{m_{n}}}$ ${\displaystyle \liminf _{n\rightarrow \infty }{\frac {m_{n+1}}{m_{n}}}>1}$

Hadamard's set of gaps

The gap set of Hadamard (in English-speaking countries also Ostrowski Hadamard Gap theorem called) now makes the statement that means Hadamardscher gap rows given holomorphic functions nowhere continued analytically are.

More precisely:

Is a complex-valued power series     with the radius of convergence of   a Hadamard series gap, the corresponding holomorphic function is   nowhere on the open convergence circular disk     also be continued and the boundary   is the natural boundary. ${\displaystyle P(z)}$   ${\displaystyle R>0}$ ${\displaystyle z\mapsto P(z)}$   ${\displaystyle B_{R}=\{z\colon \,|z|<R\}\subseteq \mathbb {C} }$   ${\displaystyle \partial {B_{R}}=\{z\colon \,|z|=R\}}$

Related sentence: Fabry's gap sentence

One of the sharpest non-continuability theorems is Fabry's gap theorem - so named after the French mathematician Eugène Fabry - which even includes Hadamard's gap theorem and reads as follows:

If a complex-valued power series   with the radius of convergence   fulfills     the condition    , the associated holomorphic function can   not be   continued anywhere beyond the open disk of convergence   and the boundary     forms the natural limit. ${\displaystyle P(z)=\sum _{n=0}^{\infty }a_{n}z^{m_{n}}}$ ${\displaystyle R}$ ${\displaystyle \lim _{n\rightarrow \infty }{\frac {m_{n}}{n}}=\infty }$ ${\displaystyle z\mapsto P(z)}$   ${\displaystyle B_{R}=\{z\colon \,|z|<R\}\subseteq \mathbb {C} }$ ${\displaystyle \partial {B_{R}}=\{z\colon \,|z|=R\}}$

Summary

The statement of both sentences can be summarized as follows:

Under the respective conditions, the convergence circle is     the holomorphic area of the associated holomorphic function   ${\displaystyle B_{R}=\{z\colon \,|z|<R\}\subseteq \mathbb {C} }$ ${\displaystyle z\mapsto P(z)}$

Two examples

(1) ${\displaystyle P(z)=\sum _{n=0}^{\infty }z^{2^{n}}}$

(2) ${\displaystyle P(z)=\sum _{n=0}^{\infty }z^{n!}}$

In both cases the holomorphic area is the unit disk .

literature

• Heinrich Behnke , Friedrich Sommer : Theory of the analytical functions of a complex variable (=  The basic teachings of the mathematical sciences in individual representations . Volume 77 ). Springer-Verlag, Berlin / Göttingen / Heidelberg 1965.
• Alexander Dinghas : Lectures on Function Theory (=  The Basic Teachings of Mathematical Sciences in Individual Representations . Volume 110 ). Springer-Verlag, Berlin / Göttingen / Heidelberg 1961. MR0179329
• J. Hadamard : Essai sur l'étude des fonctions données par leur développement de Taylor . In: Journ. Pure math. appl. tape 8 , 1892, p. 101-186 .
• J.-P. Kahane : Lacunary Taylor and Fourier series . In: Bull. Amer. Math. Soc . tape 70 , 1964, pp. 199-213 . MR0162919
• 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 .
• Walter Rudin : Real and Complex Analysis . 2nd improved edition. Oldenbourg Wissenschaftsverlag, Munich 2009, ISBN 978-3-486-59186-6 .

References and comments

1. ↑ 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 , pp. 249 .
2. ↑ In English-language sources one speaks of the lacunary series ; see. for example the work of Kahane, in: Bull. Amer. Math. Soc. , Volume 70, p. 199 ff. Or Lacunary series in the English language Wikipedia.
3. ↑ 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 , pp. 241 ff .
4. ^ Walter Rudin : Real and Complex Analysis . 2nd improved edition. Oldenbourg Wissenschaftsverlag, Munich 2009, ISBN 978-3-486-59186-6 , p. 384 ff .
5. ↑ According to Remmert / Schumacher, Funktionentheorie 2 , pp. 121, 251, the discovery of natural limits goes back to Karl Weierstrass and Leopold Kronecker .
6. ↑ If you set    , you have a representation    that you can also find in the literature. ${\displaystyle b_{n}=a_{m_{n}}\;(n=0,1,2,\dots )}$ ${\displaystyle P(z)=\sum _{n=0}^{\infty }b_{n}z^{m_{n}}}$
7. ↑ See Ostrowski-Hadamard Gap Theorem at MATHWORLD. The English naming is related to the fact that Hadamard's gap sentence can be traced back to a sentence by Alexander Markowitsch Ostrowski ; s. Walter Rudin : Real and Complex Analysis . 2nd improved edition. Oldenbourg Wissenschaftsverlag, Munich 2009, ISBN 978-3-486-59186-6 , p. 384 ff .
8. ↑ Heinrich Behnke , Friedrich Sommer : Theory of the analytical functions of a complex variable (=  The basic teachings of the mathematical sciences in individual representations . Volume 77 ). Springer-Verlag, Berlin / Göttingen / Heidelberg 1965, p. 183 .
9. ↑ 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 , pp. 249 .
10. ^ Walter Rudin : Real and Complex Analysis . 2nd improved edition. Oldenbourg Wissenschaftsverlag, Munich 2009, ISBN 978-3-486-59186-6 , p. 385-386 .
11. ↑ The gap theorem is often referred to in English-language sources as Fabry's gap theorem (or similar).
12. ↑ Dinghas: Lectures on Function Theory . S. 127 ff .
13. ↑ 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 , pp. 253 .
14. ↑ The case     is not excluded in each case . ${\displaystyle B_{R}=\mathbb {C} }$
15. ↑ 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 , pp. 250 .
16. ^ Walter Rudin : Real and Complex Analysis . 2nd improved edition. Oldenbourg Wissenschaftsverlag, Munich 2009, ISBN 978-3-486-59186-6 , p. 384 .