I'm studying real analysis and I know about derivative, Riemann integral, sequence and series, basic concepts. I'm having trouble understanding the sufficient conditions for a Fourier series of a function converges to that function. Is the following statements true or false?

  1. If the function $f$ is continuous and $2\pi$-periodic then the partial sum of the fourier series of $f$ converges pointwise to $f$ for every $x$.

  2. If the function $f$ is continuous and $2\pi$-periodic with the derivative being absolutely (Riemann) integrable on $[0,2\pi]$ then the partial sum of the Fourier series of $f$ converges uniformly to $f$ on $[0,2\pi]$

  3. If the function $f$ is continuous and $2\pi$-periodic then $f$ and the fourier series satisfy the Parseval's identity:

$$\int_0^{2\pi}|f(x)|^2dx=\pi a_0^2+\sum_{i=1}^\infty\pi(a_i^2+b_i^2)$$

  1. False. There exists example for continuous function but Fourier series diverges at some point.

  2. Yes. It means $|f^{'}|$ is bounded, hence $f$ is Lipschitz on the circle ,we get uniform convergence.

  3. Yes. Mean convergence ($L^2$ convergence) only requires integrability.

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  • $\begingroup$ Can you give me some references about these? I would love to know when it's right to use fourier series, thanks! $\endgroup$ – quangtu123 Oct 18 '14 at 4:15
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    $\begingroup$ @Mr. T For Q1, you can find an example in Fourier analysis textbook written by Stein, for Q2,3 see this. In particular, theorem 1.7 and corollary 1.18. $\endgroup$ – John Oct 18 '14 at 4:32
  • $\begingroup$ In Q3, it must be $\frac 1 2 {\pi a_0^2}$ $\endgroup$ – HLong Oct 18 '14 at 10:05
  • $\begingroup$ @HLong shouldn't it be $2\pi a_0^2$? I thought it's a typo. $\endgroup$ – John Oct 18 '14 at 10:29

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