Definition talk:Sampling Function

Isn't $\operatorname {III}_T$ a map from $\R$ to the space of distributions (ie. $\R \to \map {\DD'} \R$) not a map $\map \DD \R \to \R$? I would also prefer writing $\delta_{x - T n}$. We should also show that $\operatorname {III}_T$ is well-defined. Caliburn (talk) 16:10, 24 October 2022 (UTC)

Of course, you are right. The things are:

$\paren 1 : \operatorname {III}_T : \R \to \map {\DD'} \R$

$\paren 2 : \map {\operatorname {III}_T} x : \map \DD \R \to \C$

The question is which one is called the sampling function? --Usagiop (talk) 18:43, 24 October 2022 (UTC)

$\paren 2$ should be the sampling function. --Usagiop (talk) 19:05, 24 October 2022 (UTC)

Good call, it does seem to be $\C$, as I look more closely at Bracewell. --prime mover (talk) 19:14, 24 October 2022 (UTC)

Is this really defined as a distribution, not as a tempered distribution? I think, tempered is essential, to be able to consider its Fourier transform in Sampling Function is its own Fourier Transform. --Usagiop (talk) 20:03, 24 October 2022 (UTC)

Do you have a source for a definition yourself? If so, use it. But please don't just guess. --prime mover (talk) 20:07, 24 October 2022 (UTC)

I need to search a good English reference then. The issue is a distribution cannot be Fourier transformed, generally. Of course, we can define it as a distribution first, and then prove that it is indeed a tempered distribution and therefore it can be Fourier transformed and so on. --Usagiop (talk) 20:17, 24 October 2022 (UTC)

I don't know, all I know is what I originally posted up from the Bracewell book. --prime mover (talk) 16:33, 24 October 2022 (UTC)