Convergence of Fourier series in L^{\infty}-norm - MathOverflow [closed]

Convergence of Fourier series in L^{\infty}-norm

2011-04-03T11:53:47Z

As we know, for $1<p<\infty$ , the Fourier series of $f\in L^{p}(T)$ converges to $f$ in $L^{p}$ -norm. But is there any results concerning the convergence of Fourier series in $L^{\infty}$ -norm? Since $L^{\infty}(T)$ is not separable, the trigonometric system fails to form a Schauder basis of $L^{\infty}(T)$ , this implies that the Fourier series of $L^{\infty}(T)$ -functions fails to converge in $L^{\infty}$ -norm. But does the Fourier series of $f$ converge in $L^{\infty}$ -norm for every $f\in C(T)$?

Answer by Shaoming Guo for Convergence of Fourier series in L^{\infty}-norm

2011-04-03T12:58:21Z

I think you can find the answer in P191 "Classical Fourier Analysis",second edition, Loukas Grafakos

Answer by Gian Maria Dall'Ara for Convergence of Fourier series in L^{\infty}-norm

2011-04-03T13:31:35Z

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.