Irrational series II Summation by packages
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866910045092970496 |
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| author | Thom, Olivier |
| author_facet | Thom, Olivier |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_07350 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Irrational series II Summation by packages Thom, Olivier Complex Variables 30B10 (Primary) 46F15 (Secondary) 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. |
| title | Irrational series II Summation by packages |
| topic | Complex Variables 30B10 (Primary) 46F15 (Secondary) |
| url | https://arxiv.org/abs/2603.07350 |