Editing Discretization (section)

== Discretization of smooth functions ==

{{Main|Distribution (mathematics)#Convolution_versus_Multiplication}}
In [[generalized function]]s theory, '''discretization'''
arises as a particular case of the [[Convolution theorem#Convolution theorem for tempered distributions|Convolution Theorem]]
on [[Distribution (mathematics)#Convolution versus multiplication|tempered distributions]]

: <math>\mathcal{F}\{f*\operatorname{III}\} = \mathcal{F}\{f\} \cdot \operatorname{III}</math>
: <math>\mathcal{F}\{\alpha \cdot \operatorname{III}\}= \mathcal{F}\{\alpha\}*\operatorname{III}</math>

where <math>\operatorname{III}</math> is the [[Dirac comb]],
<math>\cdot \operatorname{III}</math> is discretization, <math>* \operatorname{III}</math> is
[[Periodic summation|periodization]], <math>f</math> is a rapidly decreasing tempered distribution
(e.g. a [[Dirac delta function]] <math>\delta</math> or any other
[[Support (mathematics)|compactly supported]] function), <math>\alpha</math> is a [[Smoothness|smooth]],
[[Distribution (mathematics)#Convolution versus multiplication|slowly growing]]
[[Function (mathematics)|ordinary function]] (e.g. the function that is constantly <math>1</math>
or any other [[Bandlimiting|band-limited]] function)
and <math>\mathcal{F}</math> is the (unitary, ordinary frequency) [[Fourier transform]].
Functions <math>\alpha</math> which are not smooth can be made smooth using a [[mollifier]] prior to discretization.

As an example, discretization of the function that is constantly <math>1</math> yields the [[sequence]] <math>[..,1,1,1,..]</math> which, interpreted as the coefficients of a [[linear combination]] of [[Dirac delta function]]s, forms a [[Dirac comb]]. If additionally [[truncation]] is applied, one obtains finite sequences, e.g. <math>[1,1,1,1]</math>. They are discrete in both, time and frequency.