Fourier Series from Eigenfunctions

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Homework Statement


"Using the eigenfunctions for the Hamiltonian of an infinite square-well potential defined over[-1,1] in the standard, dimensionless setting, construct Fourier series representation of the following functions..." the functions are e^(-100x^2), e^(-5x^2), e^(-x^2)
It also requests I find the RMS and say how many basis functions are required for a certain error but that shouldn't be too hard once I get going.

Homework Equations





The Attempt at a Solution



My question is I am a little confused as what the initial part is asking. I am confused because the eigenfunctions for the Hamiltonian are Sine and Cosine for classical mechanics but Hermite polynomials for quantum.(maybe this is wrong but I thought it was) So which ones am I supposed to use and how exactly do I construct the Fourier series from them? I know how to construct a Fourier Series in the most basic sense where you solve for the coefficients ao, an, and bn. I do know that when trying to approximate a function with the form e^(-x^2) form you end up using the error function, however, I don't think this is what the question is getting at. If someone could just help get me started with doing the Fourier Series, or maybe better, how to think about connecting this idea of using the Eigenfunctions to generating a Fourier Series. Thank you in advance.
Regards,
Caleb
PS as indicated by my name my background is mostly in Chemistry but I and doing more physics so I do have some gaps in my physics/math background which is probably why this question is confusing for me.

Homework Statement





Homework Equations





The Attempt at a Solution

 

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