How to understand the convergence of Fourier Series in $L^p$

Question

My professor told me that

Suppose that $f \in L^p(-\pi, \pi)$ (i.e. $f$ is 2$\pi$-periodic and $\|f\|_{L^p} < \infty$). If $1<p<\infty$, then the Fourier series of $f$ converges to $f$ in $L^p$.

If $p = 2$, then this statement is easy to understand since $e^{inx}, \, n = 0, \pm1,\pm 2,\dots$ form an orthonormal basis of $L^2$. Also, $L^2$ is Hilbert. Thus, $\sum\langle e^{inx}, f(x)\rangle e^{inx} = f(x)$ in $L^2$ sense (i.e. $\|\left(\sum\langle e^{inx}, f(x)\rangle e^{inx}\right) - f(x)\|_{L^2} = 0$).

But how to understand the above statement if $p \ne 2$? Then the space $L^p$ with $1<p<2$ or $2<p < \infty$is not even Hilbert. Hope someone can give a nice interpretation.

Answer 1

There are two main ways of understanding the convergence of Fourier series in $L^p$.

1. In $L^p$-norm, for $1 < p < \infty$. The proof of this result is -I think- due originally to Riesz in the 30's and although nontrivial is typically covered in graduate courses in Fourier analysis. The key point is to see that the Fourier multipliers associated with an interval $[-N,N]$ is bounded in $L^p$. You can read a proof of this in stard texts like [Do] or [Ka]. Norm convergence fails in $L^1$ and $L^\infty$ by an easy application of the uniform bound principle.

2. Almost everywhere for $1 < p < \infty$. This is the (more difficult) Carleson-Hunt theorem [Ca, Hu]. It was proven by Carleson in the case of $L^2$ and by Hunt for general $p$. It requires to prove a maximal bound for the Dirichlet kernel. This is note generally covered in graduate courses and is more of an advance topic. Almost everywhere convergence fails in $p = 1$.

[Ca]: Carleson, Lennart, On convergence and growth of partial sums of Fourier series, Acta Math. 116, 135-157 (1966). ZBL0144.06402.

[Do]: Duoandikoetxea, Javier, Fourier analysis. Transl. from the Spanish and revised by David Cruz-Uribe, Graduate Studies in Mathematics. 29. Providence, RI: American Mathematical Society (AMS). xviii, 222 p. (2001). ZBL0969.42001.

[Hu]: Carleson, Lennart, On convergence and growth of partial sums of Fourier series, Acta Math. 116, 135-157 (1966). ZBL0144.06402.

[Ka]: Katznelson, Yitzhak, An introduction to harmonic analysis. 2nd corr. ed, Dover Books on Advanced Mathematics. New York: Dover Publications, Inc. XIV, 264 p. $ 4.00 (1976). ZBL0352.43001.