Exponential type

The graph of the function in gray is $e^{-\pi z^{2}}$, the Gaussian restricted to the real axis. The Gaussian does not have exponential type, but the functions in red and blue are one sided approximations that have exponential type 2π.

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

$|F(z)|\leq A_\varepsilon e^{(\sigma+\varepsilon)|z|}$

for every $z\in\mathbb{C}$. We say F(z) is of exponential type if F(z) is of exponential type σ for some σ > 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

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

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

See also

• Paley–Wiener theorem
• Paley–Wiener space

References

• Stein, E.M. (1957), "Functions of exponential type", Ann. of Math. (2) 65: 582–592, JSTOR 1970066, MR0085342