Computability, randomness and applications / Calculabilité, hasard et leurs applications

Collection Computability, randomness and applications / Calculabilité, hasard et leurs applications

Organizer(s) Bienvenu, Laurent ; Jeandel, Emmanuel ; Porter, Christopher

Date(s) 20/06/2016 - 24/06/2016

linked URL http://conferences.cirm-math.fr/1408.html

00:00:00 / 00:00:00

2 5

Carleson's Theorem and Schnorr randomness

By Johanna Franklin

Also appears in collection : Exposés de recherche

Carleson's Theorem states that for $1 < p < \infty$, the Fourier series of a function $f$ in $L^p[-\pi,\pi]$ converges to $f$ almost everywhere. We consider this theorem in the context of computable analysis and show the following two results. (1) For a computable $p > 1$, if $f$ is a computable vector in $L^p[?\pi,\pi]$ and $t_0 \in [-\pi,\pi]$ is Schnorr random, then the Fourier series for $f$ converges at $t_0$. (2) If $t_0 \in [-\pi,\pi]$ is not Schnorr random, then there is a computable function $f : [-\pi,\pi] \rightarrow \mathbb{C}$ whose Fourier series diverges at $t_0$. This is joint work with Timothy H. McNicholl, and Jason Rute.

Information about the video

Date of recording 21/06/2016
Date of publication 06/07/2016
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers
Director(s) Guillaume Hennenfent
Format MP4

Citation data

DOI 10.24350/CIRM.V.19005403
Cite this video Franklin, Johanna (21/06/2016). Carleson's Theorem and Schnorr randomness. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19005403
URL https://dx.doi.org/10.24350/CIRM.V.19005403

Domain(s)

Bibliography

Franklin, J., McNicholl, T., Rute, J. (2016). Algorithmic randomness and Fourier analysis. <arXiv:1603.01778> - http://arxiv.org/abs/1603.01778

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