Apologies if my question is poorly phrased. I'm a computer scientist trying to teach myself about generalized functions. (Simple explanations are preferred. -- Thanks.)


One of the references I'm studying states that the space of Schwartz test functions of rapid decrease is the set of infinitely differentiable functions: $\varphi: \mathbb{R} \rightarrow \mathbb{R}$ such that for all natural numbers $n$ and $r$,


$\lim\_{x\rightarrow\pm\infty} |x^n \varphi^{(r)}(x)|$


What I would like to know is why is necessary or important for test functions to decay rapidly in this manner? i.e. faster than powers of polynomials. I'd appreciate an explanation of the intuition behind this statement and if possible a simple example.


Thanks.


EDIT: the OP is actually interested in a particular 1994 paper on "Spatial Statistics" by
Kent and Mardia,
1994 Link between kriging and thin plate splines (with J. T. Kent). In Probability, Statistics and Optimization (F. P. Kelly ed.). Wiley, New York, pp 325-339.


Both are in Statistics at Leeds,


<http://www.amsta.leeds.ac.uk/~sta6kvm/> 


<http://www.maths.leeds.ac.uk/~john/> 


<http://www.amsta.leeds.ac.uk/~sta6kvm/SpatialStatistics.html> 


Scanned article:
<http://www.gigasize.com/get.php?d=90wl2lgf49c> 


FROM THE OP: Here is motivation for my question: I'm trying to understand a paper that replaces an integral $$\int f(\omega) d\omega$$ with $$\int \frac{|\omega|^{2p + 2}}{ (1 + |\omega|^2)^{p+1}} \; f(\omega) \; d\omega$$ where $p \ge 0$ ($p = -1$ yields to the unintegrable expression) because $f(\omega)$ contains a singularity at the origin i.e. is of the form $\frac{1}{\omega^2}.$ 


LATER, ALSO FROM THE OP:
I understand some parts of the paper but not all of it. For example, I am unable to justify the equations (2.5) and (2.7). Why do they take these forms and not some other form?



Splines,  harmonic analysis,  singular integrals.
I believe I now have the answer to the question. The power of $\omega$ appear from the taylor expansion of $e^{i\omega.t\_j}$ (in section 2.3 of Kent and Mardia's paper)


Thanks.


(Apologies for the seeming bit of self promotion, but I've tagged this as the correct answer.)


