Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 4 / 30C / i / Solution

Choose the Fourier transform conventionIt maps the Schwartz space to itself. For a tempered distribution $u$, define the transform by bilinear distributional duality:For an ordinary integrable function this agrees with the first formula by interchanging its integrals. The absence of complex conjugation is the distribution-pairing convention.