---
tag: 5X8PK7H6W3
kind: concept
name: Fourier–Stieltjes transform
msc: ["42A38", "42B10", "43A25"]
sources:
  - url: https://en.wikipedia.org/wiki/Fourier_transform
---

## Summary

The Fourier–Stieltjes transform extends the [[Fourier transform|DPVM34ADZV]] from integrable functions to finite [[complex measures|NY6TYHRGMS]] 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|0V5B6D6BQH]] of a probability distribution.
[[Bochner's theorem|ZRHAG4NXHC]] describes the transforms of positive finite measures as the continuous positive-definite functions.
See the [[definition|JKCFHKT13F]].

## Explanations

### L1 human

Some objects that one wants to analyse by frequency are not functions but masses placed at points, such as a unit mass at the origin.
The Fourier–Stieltjes transform handles them by integrating the waves $e^{-2\pi i x \xi}$ against the mass distribution.
A unit mass at $0$ has the constant transform $1$: it contains every frequency equally.
A unit mass at $a$ has transform $e^{-2\pi i a \xi}$, a pure oscillation.
The name refers to the Stieltjes integral $\int e^{-2\pi i x \xi}\, dF(x)$ against a function $F$ of bounded variation, which is how the transform was first written on the line.