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table of generalized Fourier and measured groupoid transforms
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Fourier-Stieltjes transforms and measured groupoid transforms are useful generalizations of the (much simpler) Fourier transform, as concisely shown in the following table- with the same format as C. Woo's Feature on 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.
Definition 0.1 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:
\begin{equation} f(x)=\int_\mathbb{R}e^{itx}d(\alpha (t), \end{equation} 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.
| $f(t)$ |
$\F{f(t)} = \hat{f}(x)$ |
Conditions* |
Explanation |
Description |
| $c$ |
$(\sqrt{2 \pi})^{-1}c$ |
Notice on the next line the |
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overline bar placed above $t(x)$ |
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| $f(t)$ |
$\int \hat{f}(x) \overline{t(x)}dx$ |
$f(t)\in{L^1(G_l)}$ , with $G_l$ a |
Fourier-Stieltjes transform |
$\hat{f}(x)\in{C_0(\hat{G_l})}$ |
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locally compact groupoid [1]; |
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$\int $ is defined via |
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a left Haar measure on $G_l$ |
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| $\hat{m}(x)$ |
$\check{m}(t)= \int e^{itx}d\hat{m}(x)$ |
as above |
Inverse Fourier-Stieltjes |
$\check{m}(t) \in{L^1(G_l)}$ , |
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transform |
([2], [3]). |
| $\hat{m}(x)$ |
$\check{m}(t) = \int e^{itx}d\hat{m}(x)$ |
When $G_l=\mathbb{R}$ , and it exists |
This is the usual |
$\check{m}(t) \in{\mathbb{R}}$ |
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only when $\hat{m}(x)$ is |
Inverse Fourier transform |
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Lebesgue integrable on |
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the entire real axis |
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*Note the `slash hat' on $\hat{f}(x)$ and $\hat{G_l}$ .
- 1
- A. Ramsay and M. E. Walter, Fourier-Stieltjes algebras of locally compact groupoids, J. Functional Anal. 148: 314-367 (1997).
- 2
- A. L. T. Paterson, The Fourier algebra for locally compact groupoids., Preprint, (2001).
- 3
- A. L. T. Paterson, The Fourier-Stieltjes and Fourier algebras for locally compact groupoids., (2003) Free PDF file download.
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"table of generalized Fourier and measured groupoid transforms" is owned by bci1. [ full author list (2) ]
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Cross-references: inverse, left Haar measure, locally compact groupoid, line, addition, continuous, integral, bounded, bounded function, monotone increasing, interval, measurable function, positive definite, complex function, domain, entire, real axis, Lebesgue integrable, function, Fourier transform, measured groupoid, Transforms
There are 3 references to this entry.
This is version 43 of table of generalized Fourier and measured groupoid transforms, born on 2008-07-04, modified 2008-10-20.
Object id is 10739, canonical name is TableOfGeneralizedFourierAndMeasuredGroupoidTransforms.
Accessed 1480 times total.
Classification:
| AMS MSC: | 55U99 (Algebraic topology :: Applied homological algebra and category theory :: Miscellaneous) |
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Pending Errata and Addenda
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