other · Centre International de Rencontres Mathématiques · 2016

Carleson's Theorem and Schnorr randomness

Franklin, Johanna

Cited by 0Open access

Abstract

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.

Cite this paper

APA 7

Franklin, J. (2016). Carleson's Theorem and Schnorr randomness. Centre International de Rencontres Mathématiques. https://doi.org/10.24350/cirm.v.19005403

MLA 9

Franklin, Johanna. “Carleson's Theorem and Schnorr randomness.” Centre International de Rencontres Mathématiques, 2016, https://doi.org/10.24350/cirm.v.19005403.

Chicago

Franklin, Johanna. 2016. “Carleson's Theorem and Schnorr randomness.” Centre International de Rencontres Mathématiques. https://doi.org/10.24350/cirm.v.19005403.

Harvard

Franklin, J. (2016) ‘Carleson's Theorem and Schnorr randomness’, Centre International de Rencontres Mathématiques. Available at: https://doi.org/10.24350/cirm.v.19005403.

BibTeX

@misc{franklin2016carleson,
  title = {{Carleson's Theorem and Schnorr randomness}},
  author = {Franklin, Johanna},
  journal = {Centre International de Rencontres Mathématiques},
  year = {2016},
  doi = {10.24350/cirm.v.19005403}
}

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