Fourier transform of a finite measure ($\hat{\mu }(\xi )=\int e^{-ix\cdot \xi }d\mu (x)$)

The angular-frequency Fourier transform of a finite measure, or a complex measure of finite total variation, is defined by the displayed integral. It is bounded by total variation and continuous by dominated convergence. An integrable density recovers the ordinary Fourier transform of a function. A measure supported on a curved surface instead leads to a Fourier extension operator.