On the Orthorecursive Expansion of Unity
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866911474879823872 |
|---|---|
| author | Cloitre, Benoit |
| author_facet | Cloitre, Benoit |
| contents | The orthorecursive expansion of unity with respect to the system $\{x, x^2, x^3, \ldots\}$ in $L^2([0,1])$ produces a sequence of rational coefficients $(c_n)$ defined by an explicit recurrence. Kalmynin and Kosenko established the bounds $c_n = O(n^{-3/2})$ and $C_N = \sum_{k=0}^{N} c_k = O(N^{-1/2})$ through intricate $L^2$-norm arguments, but left the optimal decay rates as open problems. We prove $C_N = O_{\varepsilon}(N^{-α_1+\varepsilon})$, where $α_1 \approx 1.3465$ is the smallest real part among the zeros of a transcendental function related to the digamma function. We also improve the coefficient bound to $c_n = O(n^{-2})$. The method rests on a Tauberian transfer theorem that recasts the discrete recurrence as a Volterra integral equation, whose resolvent is smooth and amenable to Mellin analysis and contour shifting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_09645 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Orthorecursive Expansion of Unity Cloitre, Benoit Number Theory 11M45 (primary), 40E05, 44A15, 45D05 (secondary) The orthorecursive expansion of unity with respect to the system $\{x, x^2, x^3, \ldots\}$ in $L^2([0,1])$ produces a sequence of rational coefficients $(c_n)$ defined by an explicit recurrence. Kalmynin and Kosenko established the bounds $c_n = O(n^{-3/2})$ and $C_N = \sum_{k=0}^{N} c_k = O(N^{-1/2})$ through intricate $L^2$-norm arguments, but left the optimal decay rates as open problems. We prove $C_N = O_{\varepsilon}(N^{-α_1+\varepsilon})$, where $α_1 \approx 1.3465$ is the smallest real part among the zeros of a transcendental function related to the digamma function. We also improve the coefficient bound to $c_n = O(n^{-2})$. The method rests on a Tauberian transfer theorem that recasts the discrete recurrence as a Volterra integral equation, whose resolvent is smooth and amenable to Mellin analysis and contour shifting. |
| title | On the Orthorecursive Expansion of Unity |
| topic | Number Theory 11M45 (primary), 40E05, 44A15, 45D05 (secondary) |
| url | https://arxiv.org/abs/2505.09645 |