Fourier transform of a ball indicator over a local field ($\widehat{1_{a+H}}(y)=\mathrm{vol}(H)\psi (ay)1_{H^{\perp }}(y)$)

For a compact open additive subgroup $H$ of a local field, translation and additive character orthogonality give the displayed transform of a coset indicator. With the self-dual Haar measure, transforming again gives $1_{a+H}(-x)$. Every Schwartz-Bruhat function is a finite linear combination of these indicators.