generalized Fourier and measured groupoid transforms

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.



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