Expand in more localized than Fourier

[Gerenuk]

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?

 

[jasonRF]

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.

 

[Gerenuk]

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?