Intuitive reason for Fourier Series Convergence

Question

I read that Fourier Series Converges to average of left side and right side limits at Jump Discontinuities. What is the intuitive explanation for it? Is it something regarding Energy minimization?

Answer 1

Well, intuitive implies subjective, anyway here are some general considerations:

1. Where else should it go?
2. This way it generalizes correctly what happens in case of no discontinuity.
3. In case a function has a removable discontinuity (left and right limit is the same, but not equal to value at the point), it converges to the value the function "should have".

And here is the argument I like the most: we call a function odd if $f(-x)=-f(x)$ and even if $f(-x)=f(x)$. It's easy to see that:

1. Every function can be decomposed into an even and an odd part (try!)
2. Even functions only have cosines in the Fourier series, odd functions only have sines.

Suppose you have to approximate an odd function with a discontinuity at the origin. Then every Fourier term is odd (i.e. contains only sines and not cosines), and so it converges to an odd function (which is zero at the origin). If the value at the origin were any other, there would be cosines in the series, which are even components.