Periodic function and Fourier expansion

Periodic function and Fourier expansion

The theorem is that if a function f is periodic with frequency w (period a), the function can be expanded as Fourier polynomials. i.e.

If f(x+na)=f(x)

Then f(x)=\sum_{k=nK}f(k)e^{-ikx}

where K=\frac{2\pi}{a}

.

The proof actually relies on common sense.

define:

 f(k)=\int f(x)e^{-ikx}dx

One can see that both f(x) and exp(-ikx) are perodic, where average of exp(-ikx) is zero. So if the two periods are not commensurate, f(k) will be zero.

The only possible nozero f(k) occurs when kx=2npi, where k=nK.