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Irrational series I Laplace transform in a neighborhood of $-\infty$

Olivier Thom

Abstract

Discrete sums of exponentials $g(w) = \sum a_β \mathrm{e}^{βw}$ with positive exponents may converge not normally in neighborhoods $H$ of $-\infty$ which do not contain half-planes. In order to obtain a decomposition of a holomorphic function $g$ in $H$ as a sum of exponentials we study the Laplace transform in general neighborhoods of $-\infty$. We adress questions such as continuity of Laplace and inverse Laplace transformations, continuity for the operation of taking partial sums, and resummation formulas.

Irrational series I Laplace transform in a neighborhood of $-\infty$

Abstract

Discrete sums of exponentials with positive exponents may converge not normally in neighborhoods of which do not contain half-planes. In order to obtain a decomposition of a holomorphic function in as a sum of exponentials we study the Laplace transform in general neighborhoods of . We adress questions such as continuity of Laplace and inverse Laplace transformations, continuity for the operation of taking partial sums, and resummation formulas.
Paper Structure (20 sections, 29 theorems, 143 equations, 4 figures)

This paper contains 20 sections, 29 theorems, 143 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Laplace transform and hyperfunctions
  4. Results
  5. Laplace transform
  6. Definitions
  7. Laplace transform on a neighborhood of $-\infty$
  8. Inverse Laplace transform
  9. Properties of Laplace transforms
  10. Duality between $\mathcal{L}$ and $\mathcal{L}^{-1}$
  11. Example : logarithmic half-planes
  12. Example : straight half-planes
  13. Continuity properties for the Laplace transform
  14. Hyperfunctions of locally finite order
  15. Continuity of partial sums
  16. ...and 5 more sections

Key Result

Proposition 1

For any neighborhood $H\in \mathscr{N}(-\infty)$ and any bounded holomorphic function $g\in E_0(H)$, the Laplace transform $\mathcal{L}g$ is a well-defined hyperfunction $\mathcal{L}g\in H^1_{\mathbb{R}^+}(\mathbb{C},\mathscr{E})$. More explicitely, for any $\theta\in ]0,\frac{\pi}{4}[$ we can consi In particular, $\mathcal{L}g$ belongs to $\mathscr{H}_\mu(\mathbb{C})$, where $\mu(\theta) = -w_0 =

Figures (4)

  • Figure 1: integration path
  • Figure 2: domain of convergence
  • Figure 3: integration path
  • Figure 4: The setup for the integration trick

Theorems & Definitions (64)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Definition 10
  • ...and 54 more