Some Generalized Laplace Transformations

  • Erwin A. K. Brüning
Part of the International Society for Analysis, Applications and Computation book series (ISAA, volume 3)


In the summation theory for divergent power series the Laplace transform with indexk > 0 (for analytic functions of exponential size at most k > 0) and the Borel transform with index k > 0 play a prominent role. In the investigation into a more flexible summation theory we encountered two new classes of transformations: The generalized Laplace transformations and the generalized Borel transformations, both are parametrized by certain Radon probability measures on the positive half line. Here we introduce these two classes of transformations and discuss some of their basic properties. The Laplace transform (resp. the Borel transform) with index k > 0 appears as the case of a special Radon probability measure.


Asymptotic Expansion Entire Function Laplace Transformation Integration Contour Compact Neighbourhood 


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  1. [1]
    Balser, W. (1994). From Divergent Power Series to Analytic Functions. Theory and Application of Multisummable Power Series. Berlin: Lecture Notes in Mathematics No. 1582, Springer-Verlag.Google Scholar
  2. [2]
    Brüning, E. (1996). “How to reconstruct an analytic function from its asymptotic expansion?” Complex Variables, Vol. 30, 199–220.CrossRefMATHGoogle Scholar
  3. [3]
    Saitoh, S. (1994). “One Approach to some general integral transforms and its applications” Integral Transforms and Special Functions, Vol. 3, No. 1, 49–85.MathSciNetCrossRefGoogle Scholar
  4. [4]
    Sansone, G. and J. Gerretsen (1960). Lectures on the theory of a complex variable. Groningen: Walters-Nordhoff.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1999

Authors and Affiliations

  • Erwin A. K. Brüning
    • 1
  1. 1.Department of Mathematicis & Applied MathematicsUniversity of Durban-WestvilleDurbanSouth Africa

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