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We have some orthogonal system $\phi_0, \phi_1,...$.

Then we say "if the function $f$ is square-integrable then we have a fourier series...".

Question 1: why do we talk about square-integrable functions? I know that if the function is square-integrable then the fourier series converges. But I dont understand why? On which step of our theory we get this conclusion? I read several chapters on fourier analysis several times and do not understand from what we get this conclusion.

Steps:

  1. we have orthogonal system $\phi_0, \phi_1,...$.

  2. we take a function and write

$f\sim c_0\phi_0+c_1\phi_1+...$

  1. to replace the $"\sim"$ sign with "=" we need to show thet this fourier series converges.

  2. we know that if functions on our orthogonal system is continuous and if our series converges uniformaly then this fourier series is correct.

Question 2: again why do we worry about square-integrable functions? Where we need them in four steps?

Mokhmad-Salekh Khekhaev
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    The Fourier series does not converge in the pointwise sense. It converges in the norm of $L^{2}$. What is nice about square integrable functions is that you always have an expansion in this sense without any smoothness conditions on $f$. – Kavi Rama Murthy Jul 29 '21 at 09:26
  • @KaviRamaMurthy : If $f\in L^2$, then Carleson's Theorem states the Fourier series converges pointwise almost everywhere to $f$. – Disintegrating By Parts Aug 13 '21 at 03:29
  • That is true. I meant to say that that the FS need not converge at every point. – Kavi Rama Murthy Aug 13 '21 at 05:01

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