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Nachbin's theorem

Nachbin's theorem

In mathematics, in the area of complex analysis, Nachbin's theorem (named after Leopoldo Nachbin) is commonly used to establish a bound on the growth rates for an analytic function. 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. In particular, Nachbin's theorem may be used to give the domain of convergence of the generalized Borel transform, given below.

Exponential type

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

in the limit of. Here, thecomplex variablez was written asto emphasize that the limit must hold in all directions θ. Letting α stand for theinfimumof all such α, one then says that the function f is of exponential type α.
For example, let. Then one says thatis of exponential type π, since π is the smallest number that bounds the growth ofalong the imaginary axis. So, for this example,Carlson's theoremcannot apply, as it requires functions of exponential type less than π.

Ψ type

Bounding may be defined for other functions besides the exponential function. In general, a functionis a comparison function if it has a series
withfor all n, and
Comparison functions are necessarilyentire, which follows from theratio test. Ifis such a comparison function, one then says that f is of Ψ-type if there exist constants M and τ such that
as. If τ is the infimum of all such τ one says that f is of Ψ-type τ.

Nachbin's theorem states that a function f(z) with the series

is of Ψ-type τ if and only if

Borel transform

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

If f is of Ψ-type τ, then the exterior of the domain of convergence of, and all of its singular points, are contained within the disk

Furthermore, one has

where thecontour of integrationγ encircles the disk. This generalizes the usual Borel transform for exponential type, where. The integral form for the generalized Borel transform follows as well. Letbe a function whose first derivative is bounded on the interval, so that
where. Then the integral form of the generalized Borel transform is
The ordinary Borel transform is regained by setting. Note that the integral form of the Borel transform is just theLaplace transform.

Nachbin resummation

Nachbin resummation (generalized Borel transform) can be used to sum divergent series that escape to the usual Borel summation or even to solve (asymptotically) integral equations of the form:

where f(t) may or may not be of exponential growth and the kernel K(u) has aMellin transform. The solution can be obtained aswithand M(n) is the Mellin transform of K(u). An example of this is the Gram series

Fréchet space

Collections of functions of exponential typecan form acompleteuniform space, namely aFréchet space, by thetopologyinduced by the countable family ofnorms

See also

  • Divergent series

  • Borel summation

  • Euler summation

  • Cesàro summation

  • Lambert summation

  • Mittag-Leffler summation

  • Phragmén–Lindelöf principle

  • Abelian and tauberian theorems

  • Van Wijngaarden transformation

References

[1]
Citation Linkwww.encyclopediaofmath.org"Function of exponential type"
Sep 28, 2019, 8:54 PM
[2]
Citation Linkwww.encyclopediaofmath.org"Borel transform"
Sep 28, 2019, 8:54 PM
[3]
Citation Linkprespacetime.comhttp://prespacetime.com/index.php/pst/issue/view/42/showToc
Sep 28, 2019, 8:54 PM
[4]
Citation Linkwww.encyclopediaofmath.org"Function of exponential type"
Sep 28, 2019, 8:54 PM
[5]
Citation Linkwww.encyclopediaofmath.org"Borel transform"
Sep 28, 2019, 8:54 PM
[6]
Citation Linkprespacetime.comhttp://prespacetime.com/index.php/pst/issue/view/42/showToc
Sep 28, 2019, 8:54 PM
[7]
Citation Linken.wikipedia.orgThe original version of this page is from Wikipedia, you can edit the page right here on Everipedia.Text is available under the Creative Commons Attribution-ShareAlike License.Additional terms may apply.See everipedia.org/everipedia-termsfor further details.Images/media credited individually (click the icon for details).
Sep 28, 2019, 8:54 PM