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I have a question regarding the convergence of the Fourier series of a function. I am studying Fourier analysis from M.Stein and in Chapter 3 it states the theorem that if a function $f(x)$ is integrable on the circle and has a Fourier expansion, the partial sums of the Fourier series converge in the Mean Square sense:

$$\frac{1}{2\pi} \int_{0}^{2\pi}\left| f(x)-S_{n}(f)(x)\right|^{2}dx\longrightarrow 0 $$ as $n\longrightarrow \infty $.

My question is, just after this theorem the book moves on to an example of a continous function that has a divergent Fourier series at a point. If the Fourier series diverges, even at a single point, shouldn't the integral shoot off to infinity and not converge?

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If you have a Riemann-integrable (or Lebesgue-integrable) function $f$ on $[0, 2\pi ]$ and a sequence of Riemann-integrable (or Lebesgue-integrable) functions $(f_{n})_{n\in\mathbb{N}}$ on $[0, 2\pi ]$, you can consider two different types of convergence. One is to require that

$$ \lim_{n\to\infty} \frac{1}{2\pi} \int_{0}^{2\pi} |f_{n}(x) - f(x)|^{2} \, {\rm d}x = 0.$$

This first property says that the sequence $(f_{n})_{n\in\mathbb{N}}$ converges in the mean square sense to $f$. Another is to require that for each $x\in [0, 2\pi ]$, $(f_{n}(x))_{n\in\mathbb{N}}$ converges to $f(x)$. This second property says that the sequence $(f_{n})_{n\in\mathbb{N}}$ converges pointwise to $f$.

Does mean square convergence imply pointwise convergence? In general, the answer is no. In fact, there even exist examples of sequences of continuous functions which converge in the mean square sense to another continuous function, yet the sequence does not converge pointwise at a single point in the domain.

A way to make sense of this is that the value of the integral of a non-negative function does not depend so much on what it does at each point, but more on what it does outside of some suitable notion of a "small" set. As a result, it is not reasonable to expect that mean square convergence will imply pointwise convergence.

The result you have mentioned states that the partial sums of the Fourier series converge to the given function in the mean square sense. But this does not imply that the partial sums of the Fourier series converge pointwise to the given function, because it is difficult to relate mean square convergence and pointwise convergence. You need additional assumptions on your function to make such a conclusion.

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