Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Irrational series II Summation by packages

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. We study different notions of convergence for these series and in particular the intuitive notion of summation by packages. Indeed, joining in packages the terms in the sum $g(w)$ whose exponents are close together, and summing first inside each package may result in massive cancellations. We show that discrete sums $g(w)$ which are bounded in what we call logarithmic neighborhoods can always be summated by packages.

Irrational series II Summation by packages

Abstract

Discrete sums of exponentials with positive exponents may converge not normally in neighborhoods of which do not contain half-planes. We study different notions of convergence for these series and in particular the intuitive notion of summation by packages. Indeed, joining in packages the terms in the sum whose exponents are close together, and summing first inside each package may result in massive cancellations. We show that discrete sums which are bounded in what we call logarithmic neighborhoods can always be summated by packages.
Paper Structure (24 sections, 19 theorems, 59 equations)

This paper contains 24 sections, 19 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Context
  3. Results
  4. Relation with transseries
  5. Irrational series in the works of Écalle
  6. Conclusion
  7. Definitions
  8. Irrational series
  9. Convergence as a hyperfunction
  10. Evanescent summation
  11. Summation by packages
  12. Diagonal integration by parts
  13. Discrete distributions
  14. Vandermonde distributions
  15. Norms of distributions
  16. ...and 9 more sections

Key Result

Theorem 1

There exists an explicit way to associate to each formal irrational series $\hat{g}$ a sequence of evanescent partial sums $\tilde{S}_n \hat{g}$ so that whenever $g$ is an irrational series with formal development $\hat{g}$, the sequence $\tilde{S}_n\hat{g}$ converges uniformly to $g$ in a neighborh

Theorems & Definitions (42)

  • Example
  • Definition 1
  • Example
  • Definition 2
  • Theorem 1
  • Corollary 1
  • proof
  • Proposition 1
  • proof
  • Theorem 2
  • ...and 32 more