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My question has 3 (similar) parts, and is as follows:

  1. Under what conditions does $f(x)$ equal its Fourier Series for all $x$, $-L \leq x \leq L$?

  2. Under what conditions does $f(x)$ equal its Fourier sine Series for all $x$, $0 \leq x \leq L$?

  3. Under what conditions does $f(x)$ equal its Fourier cosine Series for all $x$, $0 \leq x \leq L$?

JohnColtraneisJC
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Vijay Ramanujan
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  • What do you know about $f$? Are you asking for a complete characterization of pointwise convergence for Fourier series? – Robert Israel Aug 28 '17 at 18:09
  • I think that the answer to all these is: when $f$ is continuous. When $f$ has a discontinuity, the Fourier series converges to the mean of the left and right limits. – md2perpe Aug 28 '17 at 18:55
  • We know that $f(x)$ is continuous and not much else unfortunately. I think in most cases the Fourier series converges to $f(x)$, but it is at the boundaries in particular i.e. $x = 0, L$ that most problems with convergence will arise. – Vijay Ramanujan Aug 29 '17 at 03:13
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    No, continuity is not sufficient for pointwise convergence of the Fourier series. – Robert Israel Aug 30 '17 at 00:19
  • @md2perpe Did you see Robert's comment ? The convergence of the Dirichlet kernel to the Dirac delta is messy. You need $|f(x_0)-f(x)| < C |x_0-x|^\epsilon$ (Hölder continuity) for $f \ast D_n(x_0) \to f(x_0)$ nicely. Otherwise it becomes complicated and it has good chances to diverge. If you regularize the Dirichlet kernel then yes it converges to $f(x_0)$ everywhere it is continuous. – reuns Aug 30 '17 at 01:19
  • @reuns. No, I hadn't seen the comment. Thanks for giving me a notice. Now I learnt something. Probably I've seen the calculations before, but haven't remembered more than the need of continuity. – md2perpe Aug 30 '17 at 06:12

2 Answers2

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The Fourier series of a periodic continuous function of bounded variation converges pointwise to the function. On the other hand, the Fourier series of a periodic continuous function can diverge at infinitely many points. See e.g. R.E. Edwards, "Fourier Series - A Modern Introduction", section 10.3.1.

Robert Israel
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For your first part, any function which will be periodic with period 2L can be decomposed into fourier series and will take exact value for all the point's, given it is equal at -L,L and has no discontinuances in between. If it has discontinouities, it will take the average value between the two ends(Dirichlet theorem).

Sine has the same conditions as above with the additional condition that the function should be odd.In cosine, the function should be even.

Rishabh Jain
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