Fourier Transform of (Dirac delta*f(x))

Question

I am looking for Fourier transform of $f(x)\delta(x)$. Thanks

Answer 1

For $f$ continuous by definition of the compactly supported distributions $\delta(x)$ and $f(x) \delta(x)$ $$\int_{-\infty}^\infty f(x) \delta(x) e^{-2i \pi \xi x}dx\overset{def}= \lim_{n \to \infty}\int_{-\infty}^\infty f(x) \frac{n}2 1_{|x|<1/n} e^{-2i \pi \xi x}dx=f(0)$$

Answer 2

An initial simplification :

$$\delta(x)f(x) \ \ \text{is identical to} \ \ f(0)\delta(x).$$

(see Page 4 of https://www.reed.edu/physics/faculty/wheeler/documents/Miscellaneous%20Math/Delta%20Functions/Simplified%20Dirac%20Delta.pdf with $a=0$)

Knowing that FT$(\delta(x))=1$, by linearity of the FT,

$$\text{FT}(f(0)\delta(x))=f(0).$$