PlanetPhysics/Generalized Fourier Transform

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Fourier-Stieltjes Transform

Given a positive definite, measurable function $f(x)$ on the interval $(-\infty ,\infty )$ there exists a monotone increasing, real-valued bounded function $\alpha (t)$ such that:

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

for all $x\in {\mathbb {R} }$ 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 .