Timeline for Uniform convergence of Fourier (orthonormal) expansion of series
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Jul 4, 2015 at 8:57 | comment | added | corserine | In a certain sense you are being too pessimistic here. You can always differentiate term by term in the distributional sense. This is, of course, much weaker than the classical concepts but suffices for many purposes, and so, perhaps, for yours. | |
| Jun 23, 2015 at 14:13 | comment | added | Martin Kaloe | Ah I see. Yes without uniformity one cannot do much term by term. | |
| Jun 23, 2015 at 13:46 | comment | added | paul garrett | I don't know about a definitive result for "pointwise a.e." convergence in higher dimensions. But/and such a result by itself (even as difficult as the Carleson result is in one dimension) doesn't allow us to do much with the expansion without further hypotheses (hence, my mentioning uniform pointwise and Sobolev imbedding sorts-of-things). | |
| Jun 23, 2015 at 13:41 | comment | added | Martin Kaloe | Thanks. I am only concerned about pointwise a.e. convergence, not pointwise everywhere so I think Sobolev embeddings into classes of continuous functions is not needed. From reading Theorem 1 of this note (math.uni-bielefeld.de/~tpoguntk/media/fourier_abstract.pdf), the a.e. pointwise convergence of $\sum_{1}^n (u,\varphi_k)\varphi_k(x)$ to $u(x)$ holds when working in $L^2(0,1)$. Are you saying that such a result is not true/known in higher dimensions? | |
| Jun 22, 2015 at 21:35 | history | answered | paul garrett | CC BY-SA 3.0 |