Fourier Series Uniqueness and Extensions

[Metronome]

I'm having trouble squaring this claim (youtu.be/xWa5_OXI6VM?t=127) with periodic extensions. My understanding is that if we want, for example, the Fourier series for e^x within the interval [0,L], then we can use an even periodic extension, an odd periodic extension, or a "direct" periodic extension (replicating the function's graph from 0 to L), to fill in [−L,0] and make the period 2L.

However, these yield three different Fourier Series (i.e., the even extension will have cosine terms while the odd extension will have sine terms, which won't have a phase shift and thus can't be reconciled). How is this consistent with the claim in the video?

 

[Cubist]

The claim in the video is that if two functions are the same then they will have the same Fourier transform.

I guess a more complete claim would be... if two functions are the same over the full range being considered [-L,L] then they will have the same Fourier transform (see below). That is, f(x) = g(x) for all x in the range -L < x < L

Can this be said for your three functions f_odd, f_even, and f_direct?

 

[Cubist]

I got a bit mixed up there, there is a difference between Fourier transform and Fourier series. The Fourier transform does not have a limited range. The Fourier series does. Therefore the video's claim is complete! And my statement above should be...

if two functions are the same over the full range being considered [-L,L] then they will have the same Fourier series. That is, f(x) = g(x) for all x in the range -L < x < L

 

[Cubist]

TBH I'm not completely sure about my terminology above, its a long time since I've studied Fourier transforms! But I'm fairly sure that you have to consider the whole range and not just the x>0 part.

 

[Metronome]

Perhaps I'm unclear on the purpose of periodic extensions. Aren't they just a shortcut for computing Fourier Series? If I'm asked for the Fourier Series of e^x and I use an extension to find it, shouldn't my final answer still be the unique Fourier Series of e^x?

 

[Cubist]

I might be corrected by someone who knows better than me, but...

the Fourier series calculation will assume that the function repeats continuously, so for e^x from -1 to 1 you'll be getting results based on something that looks like this...



The discontinuities seen above at -1 and 1 will generate high frequencies that you might not want in your analysis of e^x. In fact the waveform above looks kind of like a sawtooth like rather than e^x. It might be that you want to eliminate this jump.

Therefore you could choose an appropriate extension. Below is the even extension (shown in green)...



NB: This is better, but it still isn't ideal (if you want to analyse the curvy part only) because you have some sharp peaks at -1 and 1. You might later learn about a technique using a "window function" to help eliminate these.

 

[Cubist]

Also, extensions are useful if your function simply isn't defined for -ve values. But sometimes you might want to just simply shift the function left instead of extending. It depends on what you want to analyse really.

But the above should make it clear that you get different output for an even extension.

You might want to use the odd extension for a function that graphs like this one does: [math] \frac{5\sqrt{x}}{\mathrm{e}^{5x}} [/math]

 

[Cubist]

the even extension will have cosine terms while the odd extension will have sine terms

...it could be that you've been taught these extensions as a way of illustrating that all even functions have Fourier series with cos terms only (plus a possible constant), and odd functions have sin terms only.

But please check with your tutor! Like I say it's been many years since I've studied this.