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| Format: | Preprint |
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2024
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| Accès en ligne: | https://arxiv.org/abs/2412.12835 |
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| _version_ | 1866915453455040512 |
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| author | Ambrus, Gergely Gárgyán, Barnabás |
| author_facet | Ambrus, Gergely Gárgyán, Barnabás |
| contents | The Laplace--Pólya integral, defined by $J_n(r) = \frac1π\int_{-\infty}^\infty \mathrm{sinc}^n t \cos(rt) \mathrm{d} \, t$, appears in several areas of mathematics. We study this quantity by combinatorial methods; accordingly, our investigation focuses on the values at integer $r$'s. Our main result establishes a lower bound for the ratio $\frac{J_n(r+2)}{J_n(r)}$ which extends and generalises the previous estimates of Lesieur and Nicolas, and provides a natural counterpart to the upper estimate established in our previous work. We derive the statement by purely combinatorial, elementary arguments. As a corollary, we deduce that no subdiagonal central sections of the unit cube are extremal, apart from the minimal, maximal, and the main diagonal sections. We also prove several consequences for Eulerian numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_12835 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Estimates on the decay of the Laplace-Polya integral Ambrus, Gergely Gárgyán, Barnabás Metric Geometry Combinatorics Functional Analysis 26D15, 52A38, 52A40, 05A20 The Laplace--Pólya integral, defined by $J_n(r) = \frac1π\int_{-\infty}^\infty \mathrm{sinc}^n t \cos(rt) \mathrm{d} \, t$, appears in several areas of mathematics. We study this quantity by combinatorial methods; accordingly, our investigation focuses on the values at integer $r$'s. Our main result establishes a lower bound for the ratio $\frac{J_n(r+2)}{J_n(r)}$ which extends and generalises the previous estimates of Lesieur and Nicolas, and provides a natural counterpart to the upper estimate established in our previous work. We derive the statement by purely combinatorial, elementary arguments. As a corollary, we deduce that no subdiagonal central sections of the unit cube are extremal, apart from the minimal, maximal, and the main diagonal sections. We also prove several consequences for Eulerian numbers. |
| title | Estimates on the decay of the Laplace-Polya integral |
| topic | Metric Geometry Combinatorics Functional Analysis 26D15, 52A38, 52A40, 05A20 |
| url | https://arxiv.org/abs/2412.12835 |