Exponential type

In complex analysis, a branch of mathematics, an entire function function F(z) is said to be of exponential type \sigma>0 if for every \varepsilon>0 there exists a constant A_\varepsilon such that

|F(z)|\leq A_\varepsilon e^{(\sigma%2B\varepsilon)|z|}

for every z\in\mathbb{C}. We say F(z) is of exponential type if F(z) is of exponential type \sigma for some \sigma>0. The number

\tau(F)=\sigma=\displaystyle\limsup_{|z|\rightarrow\infty}|z|^{-1}\log|F(z)|

is the exponential type of F(z).

Exponential type with respect to a symmetric convex body

Stein (1957) has given a generalization of exponential type for entire functions of several complex variables. Suppose K is a convex, compact, and symmetric subset of \mathbb{R}^n. It is known that for every such K there is an associated norm \|\cdot\|_K with the property that

K=\{x\in\mathbb{R}^n�: \|x\|_K \leq1\}.

In other words, K is the unit ball in \mathbb{R}^{n} with respect to \|\cdot\|_K. The set

K^{*}=\{y\in\mathbb{R}^{n}:x\cdot y \leq 1 \text{ for all }x\in{K}\}

is called the polar set and is also a convex, compact, and symmetric subset of \mathbb{R}^n. Furthermore, we can write

\|x\|_K = \displaystyle\sup_{y\in K^{*}}|x\cdot y|.

We extend \|\cdot\|_K from \mathbb{R}^n to \mathbb{C}^n by

\|z\|_K = \displaystyle\sup_{y\in K^{*}}|z\cdot y|.

An entire function F(z) of n-complex variables is said to be of exponential type with respect to K if for every \varepsilon>0 there exists a constant A_\varepsilon such that

|F(z)|<A_\varepsilon e^{2\pi(1%2B\varepsilon)\|z\|_K}

for all z\in\mathbb{C}^{n}.

See also

References