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.