How to calculate this Fourier inverse?

Question

$$ F(t) = (1-t^{2})^{3} \mathbf{1}_{\{|t|<1\}}(t) $$ Then $$ \frac{1}{2\pi} \int F(t) e^{-itx} dx = ? $$

Thanks!

Answer 1

If I have understood your notation correctly, in Mathematica I get $$ f(x)=\frac{96}{x^7}\left((x^2-15)x \cos(x) + 3(5-2x^2)\sin(x) \right) $$ you can change in bounds on the inverse transform to be between $-1$ and $1$, because of the limited domain. $$ f(x) = \frac{1}{2\pi}\int_{-1}^1 (1-t^2)^3 \exp(i t x)\;dt $$ note that the constant of $2 \pi$ can vary on the definition used, and the inverse transform has the opposite sign in the exponential to the forward transform.