Modulation property of the Fourier transform ($\widehat{e^{ia\cdot x}d\nu }(\xi )=\hat{\nu }(\xi -a)$)

Multiplication by a unit-modulus linear phase shifts the Fourier transform. Substituting $e^{ia\cdot x}$ into the defining integral gives the displayed identity. For a surface measure this permits independent translations of cap transforms without changing their input norms or angular supports.