# Fourier Transform Properties Established

I have a function, f, in $L_1(R)$. I need to establish the following fourier transforms and I don't know how to do so. Can anyone guide me through one of these that would then help me get the other ones?

1. If $g(x) = f(x + h)$ then $\hat g(\xi) = \hat f(\xi)e^{2\pi ih\xi}$ for any $h \in R$

2. If $g(x) = f(\delta x)$ then $\hat g(\xi) = \delta^{−1} \hat f (\delta^{−1} \xi)$ for any $\delta > 0$.

• Try writing down the definition of $\hat f(x)$, $\hat f(x+h)$ and $\hat f(\delta x)$ for a start? PS: the Latex for "hat" is ... \hat{.} – Alexandre Halm Mar 26 '15 at 13:39
• @AlexHalm can you point me in a direction where I can find these definitions? Unfortunately, I do not have a fourier textbook. – samsonite Mar 26 '15 at 14:02
• Wikipedia ? Google ? – Alexandre Halm Mar 26 '15 at 14:03
• @AlexHalm I got both of these figured out. Now my question is (for my own personal curiosity), how would these change if the function was in $L_1(R^2)$ rather than $L_1(R)$? – samsonite Mar 26 '15 at 14:42