# Fourier series of functions with points of discontinuity

by jorgdv
Tags: convergence, discontinuity, fourier, series
 P: 12 If you have a function with countable discontinuities on an interval, I know that the Fourier series will converge to that function without those discontinuities. But how could you explain that formally? If the basis of the fourier series span the space L^2[a,b], that would include functions with countable point-discontinuities, right?
 Sci Advisor P: 5,935 The Fourier series for L^2 functions will converge to the function at all points of continuity and will converge to the average value at the discontinuities.
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P: 16,528
 Quote by mathman The Fourier series for L^2 functions will converge to the function at all points of continuity and will converge to the average value at the discontinuities.
Not really. There is a continuous function whose Fourier series does not converge. What you say is only true with some additional conditions, for example a Lipschitz condition or a differentiable condition.

P: 5,935

## Fourier series of functions with points of discontinuity

 Quote by micromass Not really. There is a continuous function whose Fourier series does not converge. What you say is only true with some additional conditions, for example a Lipschitz condition or a differentiable condition.
It depends on what you mean by convergence. I was talking about convergence in the L^2 norm.
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P: 16,528
 Quote by mathman It depends on what you mean by convergence. I was talking about convergence in the L^2 norm.
Then it's still wrong. The Fourier series of a function always converges to the function in the $L^2$-norm. Doesn't matter what the discontinuities are.

http://en.wikipedia.org/wiki/Riesz%E...ischer_theorem
 P: 12 What I was thinking is that in the L2 space there is an equivalence relation such that if the Lebesgue integral of the diference is 0, then they are equivalent. However, the functions in the trigonometric basis of Fourier are contained in C[a,b], and because C[a,b] is closed under addition, the infinite linear combination with real coefficients will also be contained in C[a,b]. So the Fourier series will converge to the continuous equivalent function in the L2 space. Is that right?
 Sci Advisor P: 5,935 http://en.wikipedia.org/wiki/Converg...Fourier_series Above appears to be a good summary, particularly the section on pointwise convergence.
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P: 16,528
 Quote by jorgdv What I was thinking is that in the L2 space there is an equivalence relation such that if the Lebesgue integral of the diference is 0, then they are equivalent. However, the functions in the trigonometric basis of Fourier are contained in C[a,b], and because C[a,b] is closed under addition, the infinite linear combination with real coefficients will also be contained in C[a,b]. So the Fourier series will converge to the continuous equivalent function in the L2 space. Is that right?
What do you mean with "infinite linear combination" and what do you mean with "converge". The answers to your question depend on that. There are multiple ways to interpret convergence or summation of functions.
 P: 12 With "infinite linear combination" I mean an infinite sum of elements contained in the space, in this case, scaled by real numbers (each element). And with "converge" in that context I meant pointwise.