I am solving a problem about calculating the Fourier transform of the following quadratic function:
$$f(x) = \frac{x^2 + 6x + 9}{16}$$
I tried to solve it directly by taking the transform of each part, but it was too big. Anyway, there are some rules that can help in solving it in an easy way, like the convolution theorem and the scaling property, but I can't figure it out.
Any idea how I can solve this easily?
Ler be $$\mathcal F\left\{g(x)\right\}=\hat{g}(\xi)=\int_{-\infty}^\infty g(x) e^{- i \xi x}\, \mathrm dx $$ and observe that $$\mathcal F\left\{x^n g(x)\right\}=i^n \frac{\mathrm d^n \hat{g}(\xi)}{\mathrm d\xi^n}=i^n\hat g^{(n)}(\xi)$$ So for $g(x)=1$ we have $\hat{g}(\xi)= 2\pi\delta(\xi)$ and then $$\mathcal F\left\{x^n \right\}=2\pi\, i^n \delta^{(n)}(\xi)$$ Thus for $f(x) = \frac{x^2 + 6x + 9}{16}$ $$ \mathcal F\left\{f(x)\right\}=\frac{\pi}{8}\left[-\delta''(\xi)+6\delta'(\xi)+9\delta(\xi)\right] $$