Advances in Special Functions of Fractional Calculus: Special Functions in Fractional Calculus and Their Applications in Engineering

From Abel Continuity Theorem to Paley-Wiener Theorem

Author(s): S. Yu, P. Agarwal and S. Kanemitsu * .

Pp: 112-120 (9)

DOI: 10.2174/9789815079333123010009

* (Excluding Mailing and Handling)

Abstract

In this note we reveal that the missing link among a few crucial results in analysis, Abel continuity theorem, convergence theorem on (generalized) Dirichlet series, Paley-Wiener theorem is the Laplace transform with Stieltjes integration. By this discovery, the reason why the domains of Stoltz path and of convergence look similar is made clear. Also as a natural intrinsic property of Stieltjes integral, the use of partial summation in existing proofs is elucidated. Secondly, we shall reveal that a basic part of the proof of Paley-Wiener theorem is a version of the Laplace transform.


Keywords: Laplace transform, Stieltjes integral, Abel continuity theorem, Paley-Wiener theorem, conformal mapping, 2010 MSC: 130E99, 44A10, 40A05.

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