Expand in more localized than Fourier

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The discussion explores the possibility of expanding a function f(x) using a periodic function g that is not necessarily exponential. It highlights that if g is periodic, it can be represented by a Fourier series, which typically involves sums of exponential functions. The conversation raises questions about the uniqueness of the expansion and the need for a back transformation. It suggests that if g is a combination of exponentials, it may not yield alternative solutions beyond the standard Fourier transform. Overall, the thread delves into the complexities of function expansion in relation to periodicity and uniqueness.
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Can I expand a function in
f(x)=\sum_k g_k(x\cdot k)
where g is a periodic function that is not an exponential?
So
g_k(a)=g_k(a+1)

What if there doesn't necessarily have to be a back transformation or if the expansion doesn't have to be unique?
 
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If g is a periodic function, then you can represent it with a Fourier series. So if g is not an exponential, it is a (most likely infinite) sum of exponentials.
 
So can I find a function for g? I thought if g is a combination of exponentials, then there won't be another solution rather than the usual Fourier transform?
 
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