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generalized Fourier and measured groupoid transforms (Topic)

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 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:

$\displaystyle f(x)=\int_\mathbb{R}e^{itx}d(\alpha (t),$ (0.1)

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.

FT and FT-Generalizations

$f(t)$ $\mathcal F{f(t)} = \hat{f}(x)= (2 \pi)^{-1}\int{e{(-itx)}dx}$ Conditions* Explanation Description
$e^{-t} \theta (t)$ $\mathcal F{[f(t)]}(x) = (2 \pi)^{-1}\int{\theta (t)e{(it^2x)}dx}$ from $-\infty$ to +$\infty$ From $Mathematica^{TM**}$  
$c$ $(\sqrt{2 \pi})^{-1}c$      
    Notice on the next line the overline bar ( $\overline{}$) placed above $t(x)$  
$f(t)$ $\int \hat{f}(x) \overline{t(x)}dx$ $f(t)\in{L^1(G_l)}$, with $G_l$ a Fourier-Stieltjes transform $\hat{f}(x)\in{C_0(\hat{G_l})}$
    locally compact groupoid [1];    
    $\int $ is defined via    
    a left Haar measure on $G_l$    
$\hat{m}(x)$ $\check{m}(t)= \int e^{itx}d\hat{m}(x)$ as above Inverse Fourier-Stieltjes $\check{m}(t) \in{L^1(G_l)}$,
      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 $\check{m}(t) \in{\mathbb{R}}$
    only when $\hat{m}(x)$ is Inverse Fourier transform  
    Lebesgue integrable on    
    the entire real axis    
*Note the `slash hat' on $\hat{f}(x)$ and $\hat{G_l}$; **Calculated numerically using this link to $Mathematica^{TM}$

Bibliography

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.



"generalized Fourier and measured groupoid transforms" is owned by bci1.
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See Also: determination of Fourier coefficients, table of Fourier and generalized transforms, generalized Fourier transform, table of Laplace transforms

Other names:  FT, FFT, TableOfFourierAndGeneralizedTransforms
Also defines:  Fourier transforms, Stieltjes-Fourier transforms
Keywords:  generalized Fourier transform table, measured groupoid transforms

Cross-references: Haar measure, locally compact groupoid, measurable function, domain, function, Fourier-Stieltjes transform, groupoid
There are 16 references to this object.

This is version 15 of generalized Fourier and measured groupoid transforms, born on 2009-04-05, modified 2009-04-24.
Object id is 628, canonical name is GeneralizedFourierAndMeasuredGroupoidTransforms.
Accessed 1778 times total.

Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
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