Exponential type

The graph of the function in gray is , 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 .

In complex analysis, a branch of mathematics, a holomorphic function is said to be of exponential type C if its growth is bounded by the exponential function eC|z| for some real-valued constant C 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 apply the Mellin transform, or 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 ez.

Basic ideaEdit

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

 

in the limit of  . Here, the complex variable z was written as   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  . Then one says that   is of exponential type π, since π is the smallest number that bounds the growth of   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 a theorem ultimately anchored in the theory of finite differences.

Formal definitionEdit

A holomorphic function   is said to be of exponential type   if for every   there exists a real-valued constant   such that

 

for   where  . We say   is of exponential type if   is of exponential type   for some  . The number

 

is the exponential type of  . The limit superior here means the limit of the supremum of the ratio outside a given radius as the radius goes to infinity. This is also the limit superior of the maximum of the ratio at a given radius as the radius goes to infinity. The limit superior may exist even if the maximum at radius r does not have a limit as r goes to infinity. For example, for the function

 

the value of

 

at   is asymptotic to   and thus goes to zero as n goes to infinity,[1] but F(z) is nevertheless of exponential type 1, as can be seen by looking at the points  .

Exponential type with respect to a symmetric convex bodyEdit

Stein (1957) has given a generalization of exponential type for entire functions of several complex variables. Suppose   is a convex, compact, and symmetric subset of  . It is known that for every such   there is an associated norm   with the property that

 

In other words,   is the unit ball in   with respect to  . The set

 

is called the polar set and is also a convex, compact, and symmetric subset of  . Furthermore, we can write

 

We extend   from   to   by

 

An entire function   of  -complex variables is said to be of exponential type with respect to   if for every   there exists a real-valued constant   such that

 

for all  .

Fréchet spaceEdit

Collections of functions of exponential type   can form a complete uniform space, namely a Fréchet space, by the topology induced by the countable family of norms

 

See alsoEdit

ReferencesEdit

  1. ^ In fact, even   goes to zero at   as n goes to infinity.
  • Stein, E.M. (1957), "Functions of exponential type", Ann. of Math., 2, 65: 582–592, doi:10.2307/1970066, JSTOR 1970066, MR 0085342