Editing Convolution (section)

=== Convolution theorem ===
The [[convolution theorem]] states that<ref>{{cite web |last1=Weisstein |first1=Eric W |title=From MathWorld--A Wolfram Web Resource |url=https://mathworld.wolfram.com/ConvolutionTheorem.html}}</ref>
: <math> \mathcal{F}\{f * g\} = \mathcal{F}\{f\}\cdot \mathcal{F}\{g\}</math>

where <math> \mathcal{F}\{f\}</math> denotes the [[Fourier transform]] of <math>f</math>.

==== Convolution in other types of transformations ====

Versions of this theorem also hold for the [[Laplace transform]], [[two-sided Laplace transform]], [[Z-transform]] and [[Mellin transform]].

==== Convolution on matrices ====

If <math>\mathcal W</math> is the [[DFT matrix|Fourier transform matrix]], then
: <math>\mathcal W\left(C^{(1)}x \ast C^{(2)}y\right) = \left(\mathcal W C^{(1)} \bull \mathcal W C^{(2)}\right)(x \otimes y) = \mathcal W C^{(1)}x \circ \mathcal W C^{(2)}y</math>,

where <math> \bull </math> is [[Khatri–Rao product#Face-splitting product|face-splitting product]],<ref name="slyusar">{{Cite journal|last=Slyusar|first=V. I.|date= December 27, 1996|title=End products in matrices in radar applications. |url=https://slyusar.kiev.ua/en/IZV_1998_3.pdf |archive-url=https://web.archive.org/web/20130811122444/https://slyusar.kiev.ua/en/IZV_1998_3.pdf |archive-date=2013-08-11 |url-status=live|journal=Radioelectronics and Communications Systems |volume=41 |issue=3|pages=50–53}}</ref><ref name="slyusar1">{{Cite journal|last=Slyusar|first=V. I.|date=1997-05-20|title=Analytical model of the digital antenna array on a basis of face-splitting matrix products. |url=https://slyusar.kiev.ua/ICATT97.pdf |archive-url=https://web.archive.org/web/20130811112059/https://slyusar.kiev.ua/ICATT97.pdf |archive-date=2013-08-11 |url-status=live|journal=Proc. ICATT-97, Kyiv|pages=108–109}}</ref><ref name="DIPED">{{Cite journal|last=Slyusar|first=V. I.|date=1997-09-15|title=New operations of matrices product for applications of radars|url=https://slyusar.kiev.ua/DIPED_1997.pdf |archive-url=https://web.archive.org/web/20130811113217/https://slyusar.kiev.ua/DIPED_1997.pdf |archive-date=2013-08-11 |url-status=live|journal=Proc. Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory (DIPED-97), Lviv.|pages=73–74}}</ref><ref name="slyusar2">{{Cite journal|last=Slyusar|first=V. I.|date=March 13, 1998|title=A Family of Face Products of Matrices and its Properties|url=https://slyusar.kiev.ua/FACE.pdf |archive-url=https://web.archive.org/web/20130811113935/https://slyusar.kiev.ua/FACE.pdf |archive-date=2013-08-11 |url-status=live|journal=Cybernetics and Systems Analysis C/C of Kibernetika I Sistemnyi Analiz.- 1999.|volume=35|issue=3|pages=379–384|doi=10.1007/BF02733426|s2cid=119661450}}</ref><ref name="general">{{Cite journal|last=Slyusar|first=V. I.|date=2003|title=Generalized face-products of matrices in models of digital antenna arrays with nonidentical channels|url=https://slyusar.kiev.ua/en/IZV_2003_10.pdf |archive-url=https://web.archive.org/web/20130811125643/https://slyusar.kiev.ua/en/IZV_2003_10.pdf |archive-date=2013-08-11 |url-status=live|journal=Radioelectronics and Communications Systems|volume=46|issue=10|pages=9–17}}</ref> <math> \otimes </math> denotes [[Kronecker product]], <math> \circ </math> denotes [[Hadamard product (matrices)|Hadamard product]] (this result is an evolving of [[count sketch]] properties<ref name="ninh">{{cite conference 
 | title = Fast and scalable polynomial kernels via explicit feature maps
 | last1 = Ninh | first1 = Pham
 | first2 = Rasmus | last2 = Pagh | author2-link = Rasmus Pagh
 | date = 2013
 | publisher = Association for Computing Machinery
 | conference = SIGKDD international conference on Knowledge discovery and data mining
 | doi = 10.1145/2487575.2487591
}}</ref>).

This can be generalized for appropriate matrices <math>\mathbf{A},\mathbf{B}</math>:

: <math>\mathcal W\left((\mathbf{A}x) \ast (\mathbf{B}y)\right) = \left((\mathcal W \mathbf{A}) \bull (\mathcal W \mathbf{B})\right)(x \otimes y) = (\mathcal W \mathbf{A}x) \circ (\mathcal W \mathbf{B}y)</math>
from the properties of the [[face-splitting product]].