I am given the function $f(x)=e^{-x^4}$
I am asked if it is true that $\displaystyle\int\limits^{\cssId{upper-bound-mathjax}{\infty}}_{\cssId{lower-bound-mathjax}{-\infty}} |\hat f(w)|^2\,\cssId{int-var-mathjax}{\mathrm{d}\class{main-var-unused-warning}{w}}=1$
Well I am not sure how and if I should use Plancherel theorem here, I understand that $f$ is a kind of gaussian and if I understood correctly, fourier transform of gaussian is another gaussian, which seems logic for me that this is true, but how I can actually prove it or deny it by using theorems and avoiding complicated calculations?