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.
| Original language | English |
|---|---|
| Title of host publication | Advances in Special Functions of Fractional Calculus |
| Subtitle of host publication | Special Functions in Fractional Calculus and Their Applications in Engineering |
| Publisher | Bentham Science Publishers |
| Pages | 112-120 |
| Number of pages | 9 |
| ISBN (Electronic) | 9789815079333 |
| ISBN (Print) | 9789815079340 |
| State | Published - 11 Apr 2023 |
| Externally published | Yes |
Keywords
- 2010 MSC: 130E99
- 40A05
- 44A10
- Abel continuity theorem
- Laplace transform
- Paley- Wiener theorem
- Stieltjes integral
- conformal mapping
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