Timeline for Carleson's Theorem on Manifolds

4 events

when toggle format	what		by	license	comment
Jan 21, 2014 at 16:13	comment	added	Greg Zitelli		In that case it would at least be interesting to know if there are any counterexamples.
Jan 15, 2014 at 19:43	comment	added	David E Speyer		This isn't directly an answer, but an interesting difference between $S^1$ and $S^3$ is that pointwise convergence can fail at a smooth point of the function being represented. Take $f$ to be $1$ on the northern hemisphere and $-1$ on the southern hemisphere. The eigenfunctions which are $SO(3)$ invariant are naturally indexed by the representations of $SU(2)$. Ordering them in the obvious way, the sum does not converge at the north and poles, even though $f$ is locally constant there. See sbseminar.wordpress.com/2011/02/18/a-peter-weyl-counter-example
Jan 15, 2014 at 19:14	comment	added	Mark Lewko		Note that the answer will depend on how you order the eigenfunctions. Ordering by eigenvalue is perhaps the most natural choice. In the case of the the $2$-d torus this gives spherical summation for $2$ dimensional Fourier series. Determining if pointwise convergence holds in this setting is a longstanding open problem.
Jan 15, 2014 at 17:41	history	asked	Greg Zitelli	CC BY-SA 3.0