User icurays1 - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T21:23:21Z http://mathoverflow.net/feeds/user/28028 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/112125/weakest-assumption-for-pointwise-convergence-of-fourier-series Weakest assumption for pointwise convergence of Fourier series icurays1 2012-11-11T23:02:33Z 2012-11-25T02:50:36Z <p>This should be a quick one, but so far books, my brain, and the internet have not produced a clear answer. Or maybe it's subtle and exposes a weakness in my understanding of FS!</p> <p>Suppose $f(x)=\sum_{k\in\mathbb{Z}}c_ke^{ikx}$, whereby we mean pointwise convergence. What properties must $f(x)$ then satisfy? Clearly continuity is too strong (take for example an appropriately defined square wave). $L^1[-\pi,\pi]$ seems troublesome as well, since term-by-term integration is not necessarily valid with only pointwise convergence. </p> <p>Thanks ahead for any tips!</p> http://mathoverflow.net/questions/112125/weakest-assumption-for-pointwise-convergence-of-fourier-series Comment by icurays1 icurays1 2012-11-12T02:38:48Z 2012-11-12T02:38:48Z The context of the question was that of Fourier series, but I think the restatement by @Alexandre is appropriate. http://mathoverflow.net/questions/112125/weakest-assumption-for-pointwise-convergence-of-fourier-series/112138#112138 Comment by icurays1 icurays1 2012-11-12T02:37:36Z 2012-11-12T02:37:36Z Excellent reference, thank you. It appears that the answer is not entirely straightforward, which is good (I thought about this for a while before asking).