Algorithmic randomness in harmonic analysis
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2026
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866917187251339264 |
|---|---|
| author | Franklin, Johanna N. Y. Rodriguez, Lucas E. Rojas, Diego A. |
| author_facet | Franklin, Johanna N. Y. Rodriguez, Lucas E. Rojas, Diego A. |
| contents | Within the last fifteen years, a program of establishing relationships between algorithmic randomness and almost-everywhere theorems in analysis and ergodic theory has developed. In harmonic analysis, Franklin, McNicholl, and Rute characterized Schnorr randomness using an effective version of Carleson's Theorem. We show here that, for computable $1<p<\infty$, the reals at which the Fourier series of a weakly computable vector in $L^p[-π,π]$ converges are precisely the Martin-Löf random reals. Furthermore, we show that radial limits of the Poisson integral of an $L^1(\mathbb{R})$-computable function coincide with the values of the function at exactly the Schnorr random reals and that radial limits of the Poisson integral of a weakly $L^1(\mathbb{R})$-computable function coincide with the values of the function at exactly the Martin-Löf random reals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_03239 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Algorithmic randomness in harmonic analysis Franklin, Johanna N. Y. Rodriguez, Lucas E. Rojas, Diego A. Logic Logic in Computer Science 03D32, 42A20, 31B05 Within the last fifteen years, a program of establishing relationships between algorithmic randomness and almost-everywhere theorems in analysis and ergodic theory has developed. In harmonic analysis, Franklin, McNicholl, and Rute characterized Schnorr randomness using an effective version of Carleson's Theorem. We show here that, for computable $1<p<\infty$, the reals at which the Fourier series of a weakly computable vector in $L^p[-π,π]$ converges are precisely the Martin-Löf random reals. Furthermore, we show that radial limits of the Poisson integral of an $L^1(\mathbb{R})$-computable function coincide with the values of the function at exactly the Schnorr random reals and that radial limits of the Poisson integral of a weakly $L^1(\mathbb{R})$-computable function coincide with the values of the function at exactly the Martin-Löf random reals. |
| title | Algorithmic randomness in harmonic analysis |
| topic | Logic Logic in Computer Science 03D32, 42A20, 31B05 |
| url | https://arxiv.org/abs/2601.03239 |