PlanetPhysics/Generalized Fourier Transform

\newcommand{\sqdiagram}[9]{Failed to parse (unknown function "\diagram"): {\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}} }

Fourier-Stieltjes Transform

Given a positive definite, [[../LebesgueMeasure/|measurable function]] $f(x)$ on the interval $(-\mathrm{\infty },\mathrm{\infty })$ there exists a monotone increasing, real-valued bounded [[../Bijective/|function]] $\alpha (t)$ such that:

$f(x)=\underset{ℝ}{\int }{e}^{itx}d(\alpha (t)),$

for all $x\in ℝ$ except a `small' set, that is a finite set which contains only a small number of values. When $f(x)$ is defined as above and if $\alpha (t)$ is nondecreasing and bounded then the measurable function defined by the above integral is called the Fourier-Stieltjes transform of $\alpha (t)$, and it is continuous in addition to being positive definite .