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  "body": "\n## Summary\n\nThe Fourier–Stieltjes transform extends the [[Fourier transform|DPVM34ADZV]] from integrable functions to finite [[complex measures|NY6TYHRGMS]] on $\\R^n$.\nIt 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.\nUp to a change of variable it is the [[characteristic function|0V5B6D6BQH]] of a probability distribution.\n[[Bochner's theorem|ZRHAG4NXHC]] describes the transforms of positive finite measures as the continuous positive-definite functions.\nSee the [[definition|JKCFHKT13F]].\n\n## Explanations\n\n### L1 human\n\nSome objects that one wants to analyse by frequency are not functions but masses placed at points, such as a unit mass at the origin.\nThe Fourier–Stieltjes transform handles them by integrating the waves $e^{-2\\pi i x \\xi}$ against the mass distribution.\nA unit mass at $0$ has the constant transform $1$: it contains every frequency equally.\nA unit mass at $a$ has transform $e^{-2\\pi i a \\xi}$, a pure oscillation.\nThe 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.\n",
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