Around the Fejér-Jackson inequality: Tight bounds for certain oscillatory functions via Laplace transform representations
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866917173405941760 |
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| author | Sadov, Sergey |
| author_facet | Sadov, Sergey |
| contents | The error of approximation of the $2π$-periodic sawtooth function $(π-x)/2$, $0\leq x<2π$, by its $n$-th Fourier polynomial is shown to be bounded by arccot$((2n+1)\sin(x/2))$. Related asymptotically tight inequalities with explicit constants are given for the integral of the Dirichlet kernel interpolated to non-integer values of frequency parameter and for the Taylor series remainder of the logarithmic function $\log(1-z)$ in the unit circle. The proofs are based on the Laplace transform representation of the Lerch Zeta function with $s=1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_23001 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Around the Fejér-Jackson inequality: Tight bounds for certain oscillatory functions via Laplace transform representations Sadov, Sergey Classical Analysis and ODEs 26D15, 42A10 The error of approximation of the $2π$-periodic sawtooth function $(π-x)/2$, $0\leq x<2π$, by its $n$-th Fourier polynomial is shown to be bounded by arccot$((2n+1)\sin(x/2))$. Related asymptotically tight inequalities with explicit constants are given for the integral of the Dirichlet kernel interpolated to non-integer values of frequency parameter and for the Taylor series remainder of the logarithmic function $\log(1-z)$ in the unit circle. The proofs are based on the Laplace transform representation of the Lerch Zeta function with $s=1$. |
| title | Around the Fejér-Jackson inequality: Tight bounds for certain oscillatory functions via Laplace transform representations |
| topic | Classical Analysis and ODEs 26D15, 42A10 |
| url | https://arxiv.org/abs/2512.23001 |