# I Fourier series representation

#### PainterGuy

Summary
Trying to understand the requirements for a function to be represented by a Fourier series
Hi,

A function which could be represented using Fourier series should be periodic and bounded. I'd say that the function should also integrate to zero over its period ignoring the DC component.

For many functions area from -π to 0 cancels out the area from 0 to π. For example, Fourier series representation #1 below approximates such a function.

For some functions area from -π to -π/2 cancels out area from -π/2 to 0, and then area from 0 to π/2 gets cancelled by the area from π/2 to π. For example, Fourier series representation #2 below approximates such a function.

I'm not sure if the function needs to integrate to zero following these two patterns, or it should just amount to zero without actually following any pattern of area cancellation. Could you please let me know if I have it right?

Could you represent a function something like this using Fourier series? I'm just trying to get general concept about Fourier series right. Thank you for your help.

Fourier series representations #1

Fourier series representations #2

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#### BvU

Homework Helper
I'd say that the function should also integrate to zero over its period ignoring the DC component.
That's a way of saying twice the same -- the DC component IS (proportional to) the integral over the period. And a Fourier series starts with the coefficient for $\cos(0)$ -- a constant.

 and, to answer your question: yes, your slightly pathological function can also be represented by a Fourier seeries

#### PainterGuy

Thank you!

But doesn't there exist a periodic function(s) which doesn't integrate to zero over its period? Thanks a lot for your help.

#### BvU

Homework Helper
You can add a constant to any periodic function and it remains periodic. And the only term that changes in the Fourier series is the $a_0$ term.
Your question is very strange for me .

#### PainterGuy

Thank you!

Yes, you are right it was a silly one! :)

#### RPinPA

Homework Helper
The original function doesn't need to be periodic. The more general application is that you want to represent the function on an interval [a, b], and you don't care about outside the interval.

The range of functions which can be approximated by a Fourier series on an interval is pretty broad. You can for instance have a jump discontinuity as in a step function. The series doesn't converge to f(x) at the point of discontinuity (for instance if you have a jump from 0 to 1 at x0, the series may converge to 1/2).

I don't recall the precise conditions for pointwise convergence of a Fourier series, but they're mentioned in this thread:
"The deeper fact is Carleson's theorem, which was one of the most difficult achievements in 20th century analysis, and tells us about the precise conditions for pointwise (actually, "pointwise almost everywhere") convergence of Fourier series"

"Fourier series representation"

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