I am learning about Fourier transform and I am not quite sure how to combine two transformations from a table at the moment. Could someone please explain me how I obtain the Fourier transform of $f(A x + B)$ in terms of $\hat{f}$? Thank you very much!
What is the Fourier transform of $f(A x + B)$ with respect to $\hat{f}$?
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fourier-analysis
fourier-transform
1 Answers
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Hint: Write $h(x)=f(Ax+B)$. Then $$h(x)=g(x-(-B/A))$$ where $$g(x)=f(Ax)$$ Now use the two following properties:
- If $g(x)=f(ax)$ where $a\in\mathbb{R}\backslash\{0\}$, then $$\hat{g}(\xi)=\frac{1}{|a|}\hat{f}(\xi/a)$$
- If $h(x)=g(x-a)$ where $a\in\mathbb{R}$ then $$\hat{h}(\xi)=e^{-ia\xi}\hat{g}(\xi)$$