Fourier Transform pdes

Question

I have an exam next week and I was hoping someone might be able to help me out with this question. Show that the Fourier transform of the function $f(t+a)$ is $e^{iwa}\hat{f}(w)$ . There is a list of 10 of these types of proofs in the book PDE by Walter Strauss. I just can't seem to get started with it.

Answer 1

The Fourier transform of $g(t) = f(t+a)$ is:

$$\hat{g}(w) = \int_{-\infty}^{+\infty} f(t+a) e^{-iwt}dt$$

Perform a change of variable $s = t+a$. Then:

$$\hat{g}(w)= \int_{-\infty}^{+\infty} f(s) e^{-iw(s-a)}ds = e^{iwa}\int_{-\infty}^{+\infty} f(s) e^{-iws}ds = e^{iwa}\hat{f}(w).$$