= Fourier transform of a ball indicator over a local field
{c}
{title2=$\widehat{\mathbf1_{a+H}}(y)=\operatorname{vol}(H)\psi(ay)\mathbf1_{H^\perp}(y)$}
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 $\mathbf1_{a+H}(-x)$. Every <Schwartz-Bruhat function> is a finite linear combination of these indicators.
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