Fourier–Stieltjes transform

Summary

The Fourier–Stieltjes transform extends the Fourier transform from integrable functions to finite complex measures on $\R^n$. It sends a measure $\mu$ to the function $\hat\mu(\xi) = \int e^{-2\pi i x \cdot \xi}\, d\mu(x)$, which is bounded and uniformly continuous. Up to a change of variable it is the characteristic function of a probability distribution. Bochner's theorem describes the transforms of positive finite measures as the continuous positive-definite functions. See the definition.