Nachbin's theorem

In mathematics, in the area of complex analysis, Nachbin's theorem (named after Leopoldo Nachbin) is a result used to establish bounds on the growth rates for analytic functions. In particular, Nachbin's theorem may be used to give the domain of convergence of the generalized Borel transform, also called Nachbin summation.

This article provides a brief review of growth rates, including the idea of a function of exponential type. Classification of growth rates based on type help provide a finer tool than big O or Landau notation, since a number of theorems about the analytic structure of the bounded function and its integral transforms can be stated.

A function ${\displaystyle f(z)}$ defined on the complex plane is said to be of exponential type if there exist constants ${\displaystyle M}$ and ${\displaystyle \alpha }$ such that

${\displaystyle |f(re^{i\theta })|\leq Me^{\alpha r}}$

in the limit of $\displaystyle r\to \infty }$. Here, the complex variable${\displaystyle z}$ was written as ${\displaystyle z=re^{i\theta }}$ to emphasize that the limit must hold in all directions ${\displaystyle \theta }$. Letting ${\displaystyle \alpha }$ stand for the infimum of all such ${\displaystyle \alpha }$, one then says that the function ${\displaystyle f}$ is of exponential type ${\displaystyle \alpha }$.

For example, let ${\displaystyle f(z)=\sin(\pi z)}$. Then one says that ${\displaystyle \sin(\pi z)}$ is of exponential type ${\displaystyle \pi }$, since ${\displaystyle \pi }$ is the smallest number that bounds the growth of ${\displaystyle \sin(\pi z)}$ along the imaginary axis. So, for this example, Carlson's theorem cannot apply, as it requires functions of exponential type less than ${\displaystyle \pi }$.

Additional function types may be defined for other bounding functions besides the exponential function. In general, a function ${\displaystyle \Psi (t)}$ is a comparison function if it has a series

$\displaystyle \Psi (t)=\sum _{n=0}^{\infty }\Psi _{n}t^{n}}$

with ${\displaystyle \Psi _{n}>0}$ for all ${\displaystyle n}$, and

${\displaystyle \lim _{n\to \infty }{\frac {\Psi _{n+1}}{\Psi _{n}}}=0.}$

Comparison functions are necessarily entire, which follows from the ratio test. If ${\displaystyle \Psi (t)}$ is such a comparison function, one then says that ${\displaystyle f}$ is of ${\displaystyle \Psi }$-type if there exist constants ${\displaystyle M}$ and $\displaystyle \tau }$ such that

${\displaystyle \left|f\left(re^{i\theta }\right)\right|\leq M\Psi (\tau r)}$

as $\displaystyle r\to \infty }$. If $\displaystyle \tau }$ is the infimum of all such $\displaystyle \tau }$ one says that ${\displaystyle f}$ is of ${\displaystyle \Psi }$-type $\displaystyle \tau }$.

Nachbin's theorem states that a function ${\displaystyle f(z)}$ with the series

${\displaystyle f(z)=\sum _{n=0}^{\infty }f_{n}z^{n}}$

is of ${\displaystyle \Psi }$-type $\displaystyle \tau }$ if and only if

$\displaystyle \limsup _{n\to \infty }\left|{\frac {f_{n}}{\Psi _{n}}}\right|^{1/n}=\tau .}$

This is naturally connected to the root test and can be considered a relative of the Cauchy–Hadamard theorem.

Nachbin's theorem has immediate applications in Cauchy theorem-like situations, and for integral transforms. For example, the generalized Borel transform is given by

${\displaystyle F(w)=\sum _{n=0}^{\infty }{\frac {f_{n}}{\Psi _{n}w^{n+1}}}.}$

If ${\displaystyle f}$ is of ${\displaystyle \Psi }$-type $\displaystyle \tau }$, then the exterior of the domain of convergence of ${\displaystyle F(w)}$, and all of its singular points, are contained within the disk

${\displaystyle |w|\leq \tau .}$

Furthermore, one has

${\displaystyle f(z)={\frac {1}{2\pi i}}\oint _{\gamma }\Psi (zw)F(w)\,dw}$

積分範囲γが円盤を囲む$\displaystyle |w|\leq \tau }$これは、指数型の関数に対する通常のボレル変換を一般化したもので、$\displaystyle \Psi (t)=e^{t}}$一般化ボレル変換の積分形式も同様に導かれる。$\displaystyle \alpha (t)$区間上でその1階導関数が有界である関数とする。${\displaystyle [0,\infty )}$そしてそれは定義方程式を満たす

${\displaystyle {\frac {1}{\Psi _{n}}}=\int _{0}^{\infty }t^{n}\,d\alpha (t)}$

どこ${\displaystyle d\alpha (t)=\alpha ^{\prime }(t)\,dt}$すると、一般化ボレル変換の積分形式は次のようになる。

${\displaystyle F(w)={\frac {1}{w}}\int _{0}^{\infty }f\left({\frac {t}{w}}\right)\,d\alpha (t).}$

通常のボレル変換は、次のように設定することで再び得られる。$\displaystyle \alpha (t)=-e^{-t}}$ボレル変換の積分形式はラプラス変換であることに注意してください。

ナハビン総和法は、ボレル総和法では扱えない発散級数を総和するために使用でき、例えば、次のような形式の積分方程式を漸近的に解くのに利用できる。

${\displaystyle g(s)=s\int _{0}^{\infty }K(st)f(t)\,dt}$

どこ${\textstyle g(s)=\sum _{n=0}^{\infty }a_{n}s^{-n}}$、${\displaystyle f(t)}$指数型である場合もそうでない場合もあり、カーネルは${\displaystyle K(u)}$メリン変換を持つ。解はナッハビン和を用いて次のように得られる。${\displaystyle f(x)=\sum _{n=0}^{\infty }{\frac {a_{n}}{M(n+1)}}x^{n}}$と共に${\displaystyle a_{n}}$から${\displaystyle g(s)}$そして${\displaystyle M(n)}$メリン変換${\displaystyle K(u)}$グラム級数はその一例である。 ${\displaystyle \pi (x)\approx 1+\sum _{n=1}^{\infty }{\frac {\log ^{n}(x)}{n\cdot n!\zeta (n+1)}}.}$

場合によっては追加条件として${\displaystyle \int _{0}^{\infty }K(t)t^{n}\,dt}$有限かつ非ゼロであること${\displaystyle n=0,1,2,3,....}$

指数型関数の集合$\displaystyle \tau }$可算ノルム族によって誘導される位相によって、完全な均一空間、すなわちフレシェ空間を形成することができる。

${\displaystyle \|f\|_{n}=\sup _{z\in \mathbb {C} }\exp \left[-\left(\tau +{\frac {1}{n}}\right)|z|\right]|f(z)|.}$

• ダイバージェントシリーズ
• ボレル総和
• オイラー総和
• チェザロ総括
• ランバート総和
• ミッタッグ＝レフラー総和
• フラグメン・リンデロフの原理
• アーベル定理とタウバー定理
• ファン・ワインガルデン変換

• L. Nachbin、「有限指数型の積分関数の概念の拡張」、Anais Acad. Brasil. Ciencias. 16 (1944) 143 – 147。
• ラルフ・P・ボアス・ジュニアおよびR・クレイトン・バック著『解析関数の多項式展開（第2版、訂正版）』（1964年）、アカデミック・プレス社（ニューヨーク）、シュプリンガー・フェルラーク社（ベルリン）。米国議会図書館カード番号63-23263。（ナッハビンの定理の記述と証明、およびこのトピックに関する概説を提供する。）
• A.F. レオンチェフ (2001) [1994]、「指数型関数」、数学百科事典、EMS Press
• A.F. レオンチェフ (2001) [1994]、「ボレル変換」、数学百科事典、EMS Press