Carleson's Theorem and Schnorr randomness
Apparaît également dans la collection : Computability, randomness and applications / Calculabilité, hasard et leurs applications
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.
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