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 $2\pi$.

In complex analysis, a branch of mathematics, a holomorphic function is said to be of exponential type if its growth is bounded by the exponential function e^z as |z|→∞. When a function is bounded in this way, it is then possible to express it as certain kinds of convergent summations over a series of other complex functions, as well as understanding when it is possible to apply techniques such as Borel summation, or, for example, to perform approximations using the Euler-MacLaurin formula. The general case is handled by Nachbin's theorem, which defines the analogous notion of Ψ-type for a general function Ψ(z) as opposed to e^z.

Basic idea

A function f(z) defined on the complex plane is said to be of exponential type if there exist constants M and τ such that

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

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

For example, let $f(z)=\sin(\pi z)$. Then one says that $\sin(\pi z)$ is of exponential type π, since π is the smallest number that bounds the growth of $\sin(\pi z)$ along the imaginary axis. So, for this example, Carlson's theorem cannot apply, as it requires functions of exponential type less than π. Similarly, the Euler-MacLaurin formula cannot be applied either, as it, too, expresses an theorem ultimately anchored in the theory of finite differences.

Formal definition

An holomorphic 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

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

for $|z|\to \infty$ where $z\in \mathbb {C}$. 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

$K=\{x\in \mathbb {R} ^{n}:\|x\|_{K}\leq 1\}$.

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, MR 0085342