This is related to waves & optics.
I am given the fourier transform of a function (the spectrum of frequences for a pulse of sound)
$$\hat f(w) = sinc((w-w_0)\tau ) $$
And now, a filter is attached so that it only allows waves with frequency $w_0$ to passthrough, and am asked to find what f(t) is like after we put the filter.
The question hints at using Fourier series to solve, but I am unaware of how I can use the Fourier transform along with the series to solve. (or any relations between them).
The solution gives that f(t) written as a fourier series is:
$$f(t) = \sum \hat f(w)cos(wt)$$
i.e. that the transform is the coefficients for this series. Why is this true?