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Small Nitpick: You cannot reconstruct a square wave even if you use infinite fourier coefficients due to Gibbs phenomenon.(https://en.m.wikipedia.org/wiki/Gibbs

Small Nitpick: You cannot reconstruct a square wave even if you use infinite fourier coefficients due to Gibbs phenomenon.(https://en.m.wikipedia.org/wiki/Gibbs_phenomenon https://en.m.wikipedia.org/wiki/Gibbs_phenomenon)

enthdegree 7y ago
sure you can, up to a negligible set
ajross 7y ago
Meta nitpick: the Gibbs phenomenon is a statement about any arbitrary finite decomposition. The limit of the wave form as the number of terms goes to infinity is in fact exactly a square wave.
- kragen 7y ago
  As usual, infinity complicates things; the series has pointwise convergence but not uniform convergence in such cases.
seiferteric 7y ago
Your link seems to contradict you? > Informally, the Gibbs phenomenon reflects the difficulty inherent in approximating a discontinuous function by a finite series of continuous sine and cosine waves. It is important to put emphasis on the word finite because even though every partial sum of the Fourier series overshoots the function it is approximating, the limit of the partial sums does not. The value of x where the maximum overshoot is achieved moves closer and closer to the discontinuity as the number of terms summed increases so, again informally, once the overshoot has passed by a particular x, convergence at that value of x is possible. > There is no contradiction in the overshoot converging to a non-zero amount, but the limit of the partial sums having no overshoot, because the location of that overshoot moves. We have pointwise convergence, but not uniform convergence. For a piecewise C1 function the Fourier series converges to the function at every point except at the jump discontinuities. At the jump discontinuities themselves the limit will converge to the average of the values of the function on either side of the jump. This is a consequence of the Dirichlet theorem.[11]
- mlevental 7y ago
  the second paragraph spells it out? at the jump the series converges to the average rather than either value.
joppy 7y ago
Doesn’t it still converge in the L2 norm? (The L2 distance between two functions is the area trapped between them).

enthdegree 7y ago

sure you can, up to a negligible set

ajross 7y ago

Meta nitpick: the Gibbs phenomenon is a statement about any arbitrary finite decomposition. The limit of the wave form as the number of terms goes to infinity is in fact exactly a square wave.

kragen 7y ago
As usual, infinity complicates things; the series has pointwise convergence but not uniform convergence in such cases.

seiferteric 7y ago

Your link seems to contradict you? > Informally, the Gibbs phenomenon reflects the difficulty inherent in approximating a discontinuous function by a finite series of continuous sine and cosine waves. It is important to put emphasis on the word finite because even though every partial sum of the Fourier series overshoots the function it is approximating, the limit of the partial sums does not. The value of x where the maximum overshoot is achieved moves closer and closer to the discontinuity as the number of terms summed increases so, again informally, once the overshoot has passed by a particular x, convergence at that value of x is possible. > There is no contradiction in the overshoot converging to a non-zero amount, but the limit of the partial sums having no overshoot, because the location of that overshoot moves. We have pointwise convergence, but not uniform convergence. For a piecewise C1 function the Fourier series converges to the function at every point except at the jump discontinuities. At the jump discontinuities themselves the limit will converge to the average of the values of the function on either side of the jump. This is a consequence of the Dirichlet theorem.[11]

mlevental 7y ago
the second paragraph spells it out? at the jump the series converges to the average rather than either value.

joppy 7y ago

Doesn’t it still converge in the L2 norm? (The L2 distance between two functions is the area trapped between them).