Relating the Fourier transform of two functions.

Question

We are given that $f\in L^1(\mathbb{R}^k)$ and that $A$ is a linear operator on $\mathbb{R}^k$ (I'm assuming that $A:\mathbb{R}^k\to\mathbb{R}^1$, but correct me if that is incorrect). We also have that $g(x)=f(Ax)$ and are supposed to tell how their Fourier transforms ($\hat{f}$ and $\hat{g}$) are related. I'm just not even sure how to start thinking about this. Any hints, tips, or tricks would be greatly appreciated! Thank in advance!

Answer 1

The operator $A$ is from $\mathbb{R}^k$ into itself, not into $\mathbb{R}^1$. If it is invertible, the change of variable $y=Ax$ gives $$\begin{align} \hat g(\xi)=\int_{\mathbb{R}^k}f(Ax)e^{ix\cdot\xi}\,dx&=\frac{1}{|A|}\int_{\mathbb{R}^k}f(y)e^{i(A^{-1}y)\cdot\xi}\,dy\\ &=\frac{1}{|A|}\int_{\mathbb{R}^k}f(y)e^{iy\cdot(A^{-1})^t\xi}\,dx=\frac{1}{|A|}\,\hat g\bigl((A^{-1})^t\xi\bigr). \end{align}$$