Past exam of the fourier analysis course of the University of Oxford / 2023 / 3 / b / i / Solution

Splitting the Fourier transform integral at zero gives two absolutely convergent integrals:Both $g_t$ and $\widehat{g_t}$ are integrable functions. Let $h_t(x)=(t^2+x^2{)}^{-1}$, so $\widehat{g_t}=2t{h}_t$. The duality property of the Fourier transform, or Fourier inversion in tempered distributions, givesThese Fourier transforms have continuous representatives by the Riemann-Lebesgue lemma, so the identity holds at every $\xi$. Since $g_t$ is an even function,