Editing Convolution theorem (section)

==Convolution theorem for tempered distributions==

The convolution theorem extends to [[Distribution (mathematics)#Convolution versus multiplication|tempered distributions]]. 
Here, <math>v</math> is an arbitrary tempered distribution:

:<math>\begin{align}
&\mathcal{F}\{u*v\} = \mathcal{F}\{u\} \cdot \mathcal{F}\{v\}\\
&\mathcal{F}\{\alpha \cdot v\}= \mathcal{F}\{\alpha\}*\mathcal{F}\{v\}.
\end{align}</math>

But <math>u = F\{\alpha\}</math> must be "rapidly decreasing" towards <math>-\infty</math> and <math>+\infty</math> in order to guarantee the existence of both, convolution and multiplication product. Equivalently, if <math>\alpha = F^{-1}\{u\}</math> is a smooth "slowly growing" ordinary function, it guarantees the existence of both, multiplication and convolution product.<ref>{{cite book | last=Horváth | first=John | author-link=John Horvath (mathematician) | title=Topological Vector Spaces and Distributions | publisher=Addison-Wesley Publishing Company | location=Reading, MA | year=1966}}</ref><ref>{{cite book | last=Barros-Neto | first=José | title=An Introduction to the Theory of Distributions | publisher=Dekker | location=New York, NY | year=1973}}</ref><ref>{{cite book | last=Petersen | first=Bent E. | title=Introduction to the Fourier Transform and Pseudo-Differential Operators | publisher=Pitman Publishing | location=Boston, MA | year=1983}}</ref>

In particular, every compactly supported tempered distribution, such as the [[Dirac delta function|Dirac delta]], is "rapidly decreasing". Equivalently, [[Bandlimiting|bandlimited functions]], such as the function that is constantly <math>1</math> are smooth "slowly growing" ordinary functions. If, for example, <math>v\equiv\operatorname{\text{Ш}}</math> is the [[Dirac comb]] both equations yield the [[Poisson summation formula]] and if, furthermore, <math>u\equiv\delta</math> is the Dirac delta then <math>\alpha \equiv 1</math> is constantly one and these equations yield the [[Dirac comb#Dirac-comb identity|Dirac comb identity]].