Convergence of Fourier series in L^{\infty}-norm - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-26T06:57:27Z http://mathoverflow.net/feeds/question/60427 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/60427/convergence-of-fourier-series-in-l-infty-norm Convergence of Fourier series in L^{\infty}-norm Acky 2011-04-03T11:53:47Z 2011-04-03T13:44:43Z <p>As we know, for <code>$1&lt;p&lt;\infty$</code>, the Fourier series of <code>$f\in L^{p}(T)$</code> converges to <code>$f$</code> in <code>$L^{p}$</code>-norm. But is there any results concerning the convergence of Fourier series in <code>$L^{\infty}$</code>-norm? Since <code>$L^{\infty}(T)$</code> is not separable, the trigonometric system fails to form a Schauder basis of <code>$L^{\infty}(T)$</code>, this implies that the Fourier series of <code>$L^{\infty}(T)$</code>-functions fails to converge in <code>$L^{\infty}$</code>-norm. But does the Fourier series of <code>$f$</code> converge in <code>$L^{\infty}$</code>-norm for every $f\in C(T)$?</p> http://mathoverflow.net/questions/60427/convergence-of-fourier-series-in-l-infty-norm/60432#60432 Answer by Shaoming Guo for Convergence of Fourier series in L^{\infty}-norm Shaoming Guo 2011-04-03T12:58:21Z 2011-04-03T12:58:21Z <p>I think you can find the answer in P191 "Classical Fourier Analysis",second edition, Loukas Grafakos</p> http://mathoverflow.net/questions/60427/convergence-of-fourier-series-in-l-infty-norm/60435#60435 Answer by Gian Maria Dall'Ara for Convergence of Fourier series in L^{\infty}-norm Gian Maria Dall'Ara 2011-04-03T13:31:35Z 2011-04-03T13:44:43Z <p>If Fourier series of continuous functions would converge in $L^\infty$, then, by the Uniform Boundedness Principle, the operator norms in $C(\mathbb{T})$ of the partial Fourier series operators $S_Nf(t):=\sum_{n=-N}^N\hat{f}(n)e^{int}$ would be uniformly bounded. You can find, for example in Katznelson book, a proof of the fact that such norms diverge logarithmically.</p>