Nachbin's theorem

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In mathematics, in the area of complex analysis, Nachbin's theorem (named after Leopoldo Nachbin) is a result used to establish bounds on the growth rates for analytic functions. In particular, Nachbin's theorem may be used to give the domain of convergence of the generalized Borel transform, also called Nachbin summation.

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.

Exponential type

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A function   defined on the complex plane is said to be of exponential type if there exist constants   and   such that

 

in the limit of  . Here, the complex variable   was written as   to emphasize that the limit must hold in all directions  . Letting   stand for the infimum of all such  , one then says that the function   is of exponential type  .

For example, let  . Then one says that   is of exponential type  , since   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  .

Ψ type

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Additional function types may be defined for other bounding functions besides the exponential function. In general, a function   is a comparison function if it has a series

 

with   for all  , and

 

Comparison functions are necessarily entire, which follows from the ratio test. If   is such a comparison function, one then says that   is of  -type if there exist constants   and   such that

 

as  . If   is the infimum of all such   one says that   is of  -type  .

Nachbin's theorem states that a function   with the series

 

is of  -type   if and only if

 

This is naturally connected to the root test and can be considered a relative of the Cauchy–Hadamard theorem.

Generalized Borel transform

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

 

Furthermore, one has

 

where the contour of integration γ encircles the disk  . This generalizes the usual Borel transform for functions of exponential type, where  . The integral form for the generalized Borel transform follows as well. Let   be a function whose first derivative is bounded on the interval   and that satisfies the defining equation

 

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 the Laplace transform.

Nachbin summation

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Nachbin summation can be used to sum divergent series that Borel summation does not, for instance to asymptotically solve integral equations of the form:

 

where  ,   may or may not be of exponential type, and the kernel   has a Mellin transform. The solution can be obtained using Nachbin summation as   with the   from   and with   the Mellin transform of  . An example of this is the Gram series  

In some cases as an extra condition we require   to be finite and nonzero for  

Fréchet space

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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 also

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References

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  • L. Nachbin, "An extension of the notion of integral functions of the finite exponential type", Anais Acad. Brasil. Ciencias. 16 (1944) 143–147.
  • Ralph P. Boas, Jr. and R. Creighton Buck, Polynomial Expansions of Analytic Functions (Second Printing Corrected), (1964) Academic Press Inc., Publishers New York, Springer-Verlag, Berlin. Library of Congress Card Number 63-23263. (Provides a statement and proof of Nachbin's theorem, as well as a general review of this topic.)
  • A.F. Leont'ev (2001) [1994], "Function of exponential type", Encyclopedia of Mathematics, EMS Press
  • A.F. Leont'ev (2001) [1994], "Borel transform", Encyclopedia of Mathematics, EMS Press