PlanetPhysics/Generalized Fourier and Measured Groupoid Transforms

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Generalized Fourier transforms
Fourier-Stieltjes transforms and measured groupoid transforms are useful generalizations of the (much simpler) Fourier transform, as concisely shown in the following table (see also [TableOfFourierTransforms Fourier transforms] ) - for the purpose of direct comparison with the latter transform. Unlike the more general Fourier-Stieltjes transform, the Fourier transform exists if and only if the function to be transformed is Lebesgue integrable over the whole real axis for $$t \in{\mathbb{R}}$$, or over the entire $${\mathbb{C}}$$ domain when $$\check{m}(t)$$ is a complex function.

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. 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.

\subsubsection*{FT and FT-Generalizations}


 * Note the 'slash hat' on $$\hat{f}(x)$$ and $$\hat{G_l}$$;
 * Calculated numerically using this link to $Mathematica^{TM}$