  This paper investigates the Fourier transform of BV functions. We obtain an
averaged Plancherel formula for BV functions and a characterisation of sets of
finite perimeter in terms of their Fourier transform. We also sharpen a result
of Herz concerning the "set derivative" of the Fourier transform of the
characteristic function of a set. Finally, we obtain a new proof of the
isoperimetric inequality and sharp bounds on the $L^2$-discrepancy of BV
functions.
