<s>
In	O
mathematics	O
(	O
in	O
particular	O
,	O
functional	B-Application
analysis	I-Application
)	O
,	O
convolution	B-Algorithm
is	O
a	O
mathematical	O
operation	O
on	O
two	O
functions	O
(	O
and	O
)	O
that	O
produces	O
a	O
third	O
function	O
(	O
)	O
that	O
expresses	O
how	O
the	O
shape	O
of	O
one	O
is	O
modified	O
by	O
the	O
other	O
.	O
</s>
<s>
The	O
term	O
convolution	B-Algorithm
refers	O
to	O
both	O
the	O
result	O
function	O
and	O
to	O
the	O
process	O
of	O
computing	O
it	O
.	O
</s>
<s>
The	O
integral	O
is	O
evaluated	O
for	O
all	O
values	O
of	O
shift	O
,	O
producing	O
the	O
convolution	B-Algorithm
function	O
.	O
</s>
<s>
Some	O
features	O
of	O
convolution	B-Algorithm
are	O
similar	O
to	O
cross-correlation	O
:	O
for	O
real-valued	O
functions	O
,	O
of	O
a	O
continuous	O
or	O
discrete	O
variable	O
,	O
convolution	B-Algorithm
(	O
)	O
differs	O
from	O
cross-correlation	O
(	O
)	O
only	O
in	O
that	O
either	O
or	O
is	O
reflected	O
about	O
the	O
y-axis	O
in	O
convolution	B-Algorithm
;	O
thus	O
it	O
is	O
a	O
cross-correlation	O
of	O
and	O
,	O
or	O
and	O
.For	O
complex-valued	O
functions	O
,	O
the	O
cross-correlation	O
operator	O
is	O
the	O
adjoint	O
of	O
the	O
convolution	B-Algorithm
operator	I-Algorithm
.	O
</s>
<s>
Convolution	B-Algorithm
has	O
applications	O
that	O
include	O
probability	O
,	O
statistics	O
,	O
acoustics	O
,	O
spectroscopy	O
,	O
signal	O
processing	O
and	O
image	B-Algorithm
processing	I-Algorithm
,	O
geophysics	O
,	O
engineering	O
,	O
physics	O
,	O
computer	B-Application
vision	I-Application
and	O
differential	O
equations	O
.	O
</s>
<s>
The	O
convolution	B-Algorithm
can	O
be	O
defined	O
for	O
functions	O
on	O
Euclidean	O
space	O
and	O
other	O
groups	O
(	O
as	O
algebraic	O
structures	O
)	O
.	O
</s>
<s>
For	O
example	O
,	O
periodic	O
functions	O
,	O
such	O
as	O
the	O
discrete-time	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
can	O
be	O
defined	O
on	O
a	O
circle	O
and	O
convolved	B-Algorithm
by	O
periodic	B-Algorithm
convolution	I-Algorithm
.	O
</s>
<s>
A	O
discrete	O
convolution	B-Algorithm
can	O
be	O
defined	O
for	O
functions	O
on	O
the	O
set	O
of	O
integers	O
.	O
</s>
<s>
Generalizations	O
of	O
convolution	B-Algorithm
have	O
applications	O
in	O
the	O
field	O
of	O
numerical	B-General_Concept
analysis	I-General_Concept
and	O
numerical	O
linear	O
algebra	O
,	O
and	O
in	O
the	O
design	O
and	O
implementation	O
of	O
finite	O
impulse	O
response	O
filters	O
in	O
signal	O
processing	O
.	O
</s>
<s>
Computing	O
the	O
inverse	O
of	O
the	O
convolution	B-Algorithm
operation	I-Algorithm
is	O
known	O
as	O
deconvolution	B-Algorithm
.	O
</s>
<s>
The	O
convolution	B-Algorithm
of	O
and	O
is	O
written	O
,	O
denoting	O
the	O
operator	O
with	O
the	O
symbol	O
.	O
</s>
<s>
As	O
such	O
,	O
it	O
is	O
a	O
particular	O
kind	O
of	O
integral	B-Algorithm
transform	I-Algorithm
:	O
</s>
<s>
At	O
each	O
t	O
,	O
the	O
convolution	B-Algorithm
formula	O
can	O
be	O
described	O
as	O
the	O
area	O
under	O
the	O
function	O
weighted	O
by	O
the	O
function	O
shifted	O
by	O
the	O
amount	O
.	O
</s>
<s>
For	O
the	O
multi-dimensional	O
formulation	O
of	O
convolution	B-Algorithm
,	O
see	O
domain	O
of	O
definition	O
(	O
below	O
)	O
.	O
</s>
<s>
respectively	O
,	O
the	O
convolution	B-Algorithm
operation	I-Algorithm
can	O
be	O
defined	O
as	O
the	O
inverse	B-Algorithm
Laplace	I-Algorithm
transform	I-Algorithm
of	O
the	O
product	O
of	O
and	O
.	O
</s>
<s>
Note	O
that	O
is	O
the	O
bilateral	B-Algorithm
Laplace	I-Algorithm
transform	I-Algorithm
of	O
.	O
</s>
<s>
A	O
similar	O
derivation	B-Algorithm
can	O
be	O
done	O
using	O
the	O
unilateral	O
Laplace	O
transform	O
(	O
one-sided	O
Laplace	O
transform	O
)	O
.	O
</s>
<s>
The	O
convolution	B-Algorithm
operation	I-Algorithm
also	O
describes	O
the	O
output	O
(	O
in	O
terms	O
of	O
the	O
input	O
)	O
of	O
an	O
important	O
class	O
of	O
operations	O
known	O
as	O
linear	O
time-invariant	B-Algorithm
(	O
LTI	O
)	O
.	O
</s>
<s>
See	O
LTI	O
system	O
theory	O
for	O
a	O
derivation	B-Algorithm
of	O
convolution	B-Algorithm
as	O
the	O
result	O
of	O
LTI	O
constraints	O
.	O
</s>
<s>
In	O
terms	O
of	O
the	O
Fourier	B-Algorithm
transforms	I-Algorithm
of	O
the	O
input	O
and	O
output	O
of	O
an	O
LTI	O
operation	O
,	O
no	O
new	O
frequency	O
components	O
are	O
created	O
.	O
</s>
<s>
See	O
Convolution	B-Algorithm
theorem	O
for	O
a	O
derivation	B-Algorithm
of	O
that	O
property	O
of	O
convolution	B-Algorithm
.	O
</s>
<s>
Conversely	O
,	O
convolution	B-Algorithm
can	O
be	O
derived	O
as	O
the	O
inverse	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
the	O
pointwise	O
product	O
of	O
two	O
Fourier	B-Algorithm
transforms	I-Algorithm
.	O
</s>
<s>
The	O
resulting	O
waveform	O
(	O
not	O
shown	O
here	O
)	O
is	O
the	O
convolution	B-Algorithm
of	I-Algorithm
functions	I-Algorithm
and	O
.	O
</s>
<s>
center|452x452px	O
In	O
this	O
example	O
,	O
the	O
red-colored	O
"	O
pulse	O
"	O
,	O
is	O
an	O
even	O
function	O
so	O
convolution	B-Algorithm
is	O
equivalent	O
to	O
correlation	O
.	O
</s>
<s>
The	O
amount	O
of	O
yellow	O
is	O
the	O
area	O
of	O
the	O
product	O
computed	O
by	O
the	O
convolution/correlation	O
integral	O
.	O
</s>
<s>
475px	O
In	O
this	O
depiction	O
,	O
could	O
represent	O
the	O
response	O
of	O
an	O
RC	O
circuit	O
to	O
a	O
narrow	O
pulse	O
that	O
occurs	O
at	O
In	O
other	O
words	O
,	O
if	O
the	O
result	O
of	O
convolution	B-Algorithm
is	O
just	O
But	O
when	O
is	O
the	O
wider	O
pulse	O
(	O
in	O
red	O
)	O
,	O
the	O
response	O
is	O
a	O
"	O
smeared	O
"	O
version	O
of	O
It	O
begins	O
at	O
because	O
we	O
defined	O
as	O
the	O
distance	O
from	O
the	O
axis	O
to	O
the	O
center	O
of	O
the	O
wide	O
pulse	O
(	O
instead	O
of	O
the	O
leading	O
edge	O
)	O
.	O
</s>
<s>
One	O
of	O
the	O
earliest	O
uses	O
of	O
the	O
convolution	B-Algorithm
integral	I-Algorithm
appeared	O
in	O
Alembert	O
's	O
derivation	B-Algorithm
of	O
Taylor	O
's	O
theorem	O
in	O
Recherches	O
sur	O
différents	O
points	O
importants	O
du	O
système	O
du	O
monde	O
,	O
published	O
in	O
1754	O
.	O
</s>
<s>
Soon	O
thereafter	O
,	O
convolution	B-Algorithm
operations	I-Algorithm
appear	O
in	O
the	O
works	O
of	O
Pierre	O
Simon	O
Laplace	O
,	O
Jean-Baptiste	O
Joseph	O
Fourier	O
,	O
Siméon	O
Denis	O
Poisson	O
,	O
and	O
others	O
.	O
</s>
<s>
Prior	O
to	O
that	O
it	O
was	O
sometimes	O
known	O
as	O
Faltung	B-Algorithm
(	O
which	O
means	O
folding	O
in	O
German	O
)	O
,	O
composition	O
product	O
,	O
superposition	B-Algorithm
integral	I-Algorithm
,	O
and	O
Carson	B-Algorithm
's	I-Algorithm
integral	I-Algorithm
.	O
</s>
<s>
When	O
a	O
function	O
is	O
periodic	O
,	O
with	O
period	O
,	O
then	O
for	O
functions	O
,	O
,	O
such	O
that	O
exists	O
,	O
the	O
convolution	B-Algorithm
is	O
also	O
periodic	O
and	O
identical	O
to	O
:	O
</s>
<s>
The	O
summation	O
is	O
called	O
a	O
periodic	B-Algorithm
summation	I-Algorithm
of	O
the	O
function	O
.	O
</s>
<s>
When	O
is	O
a	O
periodic	B-Algorithm
summation	I-Algorithm
of	O
another	O
function	O
,	O
,	O
then	O
is	O
known	O
as	O
a	O
circular	O
or	O
cyclic	B-Algorithm
convolution	I-Algorithm
of	O
and	O
.	O
</s>
<s>
And	O
if	O
the	O
periodic	B-Algorithm
summation	I-Algorithm
above	O
is	O
replaced	O
by	O
,	O
the	O
operation	O
is	O
called	O
a	O
periodic	B-Algorithm
convolution	I-Algorithm
of	O
and	O
.	O
</s>
<s>
For	O
complex-valued	O
functions	O
defined	O
on	O
the	O
set	O
Z	O
of	O
integers	O
,	O
the	O
discrete	O
convolution	B-Algorithm
of	O
and	O
is	O
given	O
by	O
:	O
</s>
<s>
The	O
convolution	B-Algorithm
of	O
two	O
finite	O
sequences	O
is	O
defined	O
by	O
extending	O
the	O
sequences	O
to	O
finitely	O
supported	O
functions	O
on	O
the	O
set	O
of	O
integers	O
.	O
</s>
<s>
When	O
the	O
sequences	O
are	O
the	O
coefficients	O
of	O
two	O
polynomials	O
,	O
then	O
the	O
coefficients	O
of	O
the	O
ordinary	O
product	O
of	O
the	O
two	O
polynomials	O
are	O
the	O
convolution	B-Algorithm
of	O
the	O
original	O
two	O
sequences	O
.	O
</s>
<s>
When	O
a	O
function	O
is	O
periodic	O
,	O
with	O
period	O
,	O
then	O
for	O
functions	O
,	O
,	O
such	O
that	O
exists	O
,	O
the	O
convolution	B-Algorithm
is	O
also	O
periodic	O
and	O
identical	O
to	O
:	O
</s>
<s>
The	O
summation	O
on	O
is	O
called	O
a	O
periodic	B-Algorithm
summation	I-Algorithm
of	O
the	O
function	O
.	O
</s>
<s>
If	O
is	O
a	O
periodic	B-Algorithm
summation	I-Algorithm
of	O
another	O
function	O
,	O
,	O
then	O
is	O
known	O
as	O
a	O
circular	B-Algorithm
convolution	I-Algorithm
of	O
and	O
.	O
</s>
<s>
The	O
notation	O
(	O
)	O
for	O
cyclic	B-Algorithm
convolution	I-Algorithm
denotes	O
convolution	B-Algorithm
over	O
the	O
cyclic	O
group	O
of	O
integers	O
modulo	O
.	O
</s>
<s>
Circular	B-Algorithm
convolution	I-Algorithm
arises	O
most	O
often	O
in	O
the	O
context	O
of	O
fast	O
convolution	B-Algorithm
with	O
a	O
fast	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
FFT	O
)	O
algorithm	O
.	O
</s>
<s>
In	O
many	O
situations	O
,	O
discrete	O
convolutions	B-Algorithm
can	O
be	O
converted	O
to	O
circular	B-Algorithm
convolutions	I-Algorithm
so	O
that	O
fast	O
transforms	O
with	O
a	O
convolution	B-Algorithm
property	O
can	O
be	O
used	O
to	O
implement	O
the	O
computation	O
.	O
</s>
<s>
For	O
example	O
,	O
convolution	B-Algorithm
of	O
digit	O
sequences	O
is	O
the	O
kernel	B-Algorithm
operation	O
in	O
multiplication	O
of	O
multi-digit	O
numbers	O
,	O
which	O
can	O
therefore	O
be	O
efficiently	O
implemented	O
with	O
transform	O
techniques	O
(	O
;	O
)	O
.	O
</s>
<s>
Digital	B-General_Concept
signal	I-General_Concept
processing	I-General_Concept
and	O
other	O
applications	O
typically	O
use	O
fast	O
convolution	B-Algorithm
algorithms	O
to	O
reduce	O
the	O
cost	O
of	O
the	O
convolution	B-Algorithm
to	O
O( log )	O
complexity	O
.	O
</s>
<s>
The	O
most	O
common	O
fast	O
convolution	B-Algorithm
algorithms	O
use	O
fast	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
FFT	O
)	O
algorithms	O
via	O
the	O
circular	B-Algorithm
convolution	I-Algorithm
theorem	O
.	O
</s>
<s>
Specifically	O
,	O
the	O
circular	B-Algorithm
convolution	I-Algorithm
of	O
two	O
finite-length	O
sequences	O
is	O
found	O
by	O
taking	O
an	O
FFT	O
of	O
each	O
sequence	O
,	O
multiplying	O
pointwise	O
,	O
and	O
then	O
performing	O
an	O
inverse	O
FFT	O
.	O
</s>
<s>
Convolutions	B-Algorithm
of	O
the	O
type	O
defined	O
above	O
are	O
then	O
efficiently	O
implemented	O
using	O
that	O
technique	O
in	O
conjunction	O
with	O
zero-extension	O
and/or	O
discarding	O
portions	O
of	O
the	O
output	O
.	O
</s>
<s>
Other	O
fast	O
convolution	B-Algorithm
algorithms	O
,	O
such	O
as	O
the	O
Schönhage	B-Algorithm
–	I-Algorithm
Strassen	I-Algorithm
algorithm	I-Algorithm
or	O
the	O
Mersenne	O
transform	O
,	O
use	O
fast	O
Fourier	B-Algorithm
transforms	I-Algorithm
in	O
other	O
rings	O
.	O
</s>
<s>
If	O
one	O
sequence	O
is	O
much	O
longer	O
than	O
the	O
other	O
,	O
zero-extension	O
of	O
the	O
shorter	O
sequence	O
and	O
fast	O
circular	B-Algorithm
convolution	I-Algorithm
is	O
not	O
the	O
most	O
computationally	O
efficient	O
method	O
available	O
.	O
</s>
<s>
Instead	O
,	O
decomposing	O
the	O
longer	O
sequence	O
into	O
blocks	O
and	O
convolving	O
each	O
block	O
allows	O
for	O
faster	O
algorithms	O
such	O
as	O
the	O
overlap	B-Algorithm
–	I-Algorithm
save	I-Algorithm
method	I-Algorithm
and	O
overlap	B-Algorithm
–	I-Algorithm
add	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
A	O
hybrid	O
convolution	B-Algorithm
method	O
that	O
combines	O
block	O
and	O
FIR	O
algorithms	O
allows	O
for	O
a	O
zero	O
input-output	O
latency	O
that	O
is	O
useful	O
for	O
real-time	O
convolution	B-Algorithm
computations	O
.	O
</s>
<s>
The	O
convolution	B-Algorithm
of	O
two	O
complex-valued	O
functions	O
on	O
is	O
itself	O
a	O
complex-valued	O
function	O
on	O
,	O
defined	O
by	O
:	O
</s>
<s>
Conditions	O
for	O
the	O
existence	O
of	O
the	O
convolution	B-Algorithm
may	O
be	O
tricky	O
,	O
since	O
a	O
blow-up	O
in	O
at	O
infinity	O
can	O
be	O
easily	O
offset	O
by	O
sufficiently	O
rapid	O
decay	O
in	O
.	O
</s>
<s>
If	O
and	O
are	O
compactly	O
supported	O
continuous	O
functions	O
,	O
then	O
their	O
convolution	B-Algorithm
exists	O
,	O
and	O
is	O
also	O
compactly	O
supported	O
and	O
continuous	O
.	O
</s>
<s>
More	O
generally	O
,	O
if	O
either	O
function	O
(	O
say	O
)	O
is	O
compactly	O
supported	O
and	O
the	O
other	O
is	O
locally	O
integrable	O
,	O
then	O
the	O
convolution	B-Algorithm
is	O
well-defined	O
and	O
continuous	O
.	O
</s>
<s>
Convolution	B-Algorithm
of	O
and	O
is	O
also	O
well	O
defined	O
when	O
both	O
functions	O
are	O
locally	O
square	O
integrable	O
on	O
and	O
supported	O
on	O
an	O
interval	O
of	O
the	O
form	O
(	O
or	O
both	O
supported	O
on	O
)	O
.	O
</s>
<s>
The	O
convolution	B-Algorithm
of	O
and	O
exists	O
if	O
and	O
are	O
both	O
Lebesgue	O
integrable	O
functions	O
in	O
(	O
)	O
,	O
and	O
in	O
this	O
case	O
is	O
also	O
integrable	O
.	O
</s>
<s>
This	O
is	O
also	O
true	O
for	O
functions	O
in	O
,	O
under	O
the	O
discrete	O
convolution	B-Algorithm
,	O
or	O
more	O
generally	O
for	O
the	O
convolution	B-Algorithm
on	O
any	O
group	O
.	O
</s>
<s>
In	O
the	O
particular	O
case	O
,	O
this	O
shows	O
that	O
is	O
a	O
Banach	O
algebra	O
under	O
the	O
convolution	B-Algorithm
(	O
and	O
equality	O
of	O
the	O
two	O
sides	O
holds	O
if	O
and	O
are	O
non-negative	O
almost	O
everywhere	O
)	O
.	O
</s>
<s>
More	O
generally	O
,	O
Young	O
's	O
inequality	O
implies	O
that	O
the	O
convolution	B-Algorithm
is	O
a	O
continuous	O
bilinear	O
map	O
between	O
suitable	O
spaces	O
.	O
</s>
<s>
so	O
that	O
the	O
convolution	B-Algorithm
is	O
a	O
continuous	O
bilinear	O
mapping	O
from	O
to	O
.	O
</s>
<s>
The	O
Young	O
inequality	O
for	O
convolution	B-Algorithm
is	O
also	O
true	O
in	O
other	O
contexts	O
(	O
circle	O
group	O
,	O
convolution	B-Algorithm
on	O
)	O
.	O
</s>
<s>
Convolution	B-Algorithm
also	O
defines	O
a	O
bilinear	O
continuous	O
map	O
for	O
,	O
owing	O
to	O
the	O
weak	O
Young	O
inequality	O
:	O
</s>
<s>
In	O
addition	O
to	O
compactly	O
supported	O
functions	O
and	O
integrable	O
functions	O
,	O
functions	O
that	O
have	O
sufficiently	O
rapid	O
decay	O
at	O
infinity	O
can	O
also	O
be	O
convolved	B-Algorithm
.	O
</s>
<s>
An	O
important	O
feature	O
of	O
the	O
convolution	B-Algorithm
is	O
that	O
if	O
f	O
and	O
g	O
both	O
decay	O
rapidly	O
,	O
then	O
f∗g	O
also	O
decays	O
rapidly	O
.	O
</s>
<s>
In	O
particular	O
,	O
if	O
f	O
and	O
g	O
are	O
rapidly	O
decreasing	O
functions	O
,	O
then	O
so	O
is	O
the	O
convolution	B-Algorithm
f∗g	O
.	O
</s>
<s>
Combined	O
with	O
the	O
fact	O
that	O
convolution	B-Algorithm
commutes	O
with	O
differentiation	O
(	O
see	O
#Properties	O
)	O
,	O
it	O
follows	O
that	O
the	O
class	O
of	O
Schwartz	B-Algorithm
functions	I-Algorithm
is	O
closed	O
under	O
convolution	B-Algorithm
.	O
</s>
<s>
This	O
agrees	O
with	O
the	O
convolution	B-Algorithm
defined	O
above	O
when	O
μ	O
and	O
ν	O
are	O
regarded	O
as	O
distributions	O
,	O
as	O
well	O
as	O
the	O
convolution	B-Algorithm
of	O
L1	O
functions	O
when	O
μ	O
and	O
ν	O
are	O
absolutely	O
continuous	O
with	O
respect	O
to	O
the	O
Lebesgue	O
measure	O
.	O
</s>
<s>
Because	O
the	O
space	O
of	O
measures	O
of	O
bounded	O
variation	O
is	O
a	O
Banach	O
space	O
,	O
convolution	B-Algorithm
of	O
measures	O
can	O
be	O
treated	O
with	O
standard	O
methods	O
of	O
functional	B-Application
analysis	I-Application
that	O
may	O
not	O
apply	O
for	O
the	O
convolution	B-Algorithm
of	O
distributions	O
.	O
</s>
<s>
The	O
convolution	B-Algorithm
defines	O
a	O
product	O
on	O
the	O
linear	O
space	O
of	O
integrable	O
functions	O
.	O
</s>
<s>
This	O
product	O
satisfies	O
the	O
following	O
algebraic	O
properties	O
,	O
which	O
formally	O
mean	O
that	O
the	O
space	O
of	O
integrable	O
functions	O
with	O
the	O
product	O
given	O
by	O
convolution	B-Algorithm
is	O
a	O
commutative	O
associative	O
algebra	O
without	O
identity	O
.	O
</s>
<s>
Other	O
linear	O
spaces	O
of	O
functions	O
,	O
such	O
as	O
the	O
space	O
of	O
continuous	O
functions	O
of	O
compact	O
support	O
,	O
are	O
closed	O
under	O
the	O
convolution	B-Algorithm
,	O
and	O
so	O
also	O
form	O
commutative	O
associative	O
algebras	O
.	O
</s>
<s>
Multiplicative	O
identity	O
No	O
algebra	O
of	O
functions	O
possesses	O
an	O
identity	O
for	O
the	O
convolution	B-Algorithm
.	O
</s>
<s>
The	O
lack	O
of	O
identity	O
is	O
typically	O
not	O
a	O
major	O
inconvenience	O
,	O
since	O
most	O
collections	O
of	O
functions	O
on	O
which	O
the	O
convolution	B-Algorithm
is	O
performed	O
can	O
be	O
convolved	B-Algorithm
with	O
a	O
delta	O
distribution	O
(	O
a	O
unitary	O
impulse	O
,	O
centered	O
at	O
zero	O
)	O
or	O
,	O
at	O
the	O
very	O
least	O
(	O
as	O
is	O
the	O
case	O
of	O
L1	O
)	O
admit	O
approximations	O
to	O
the	O
identity	O
.	O
</s>
<s>
The	O
linear	O
space	O
of	O
compactly	O
supported	O
distributions	O
does	O
,	O
however	O
,	O
admit	O
an	O
identity	O
under	O
the	O
convolution	B-Algorithm
.	O
</s>
<s>
Inverse	O
element	O
Some	O
distributions	O
S	O
have	O
an	O
inverse	O
element	O
S−1	O
for	O
the	O
convolution	B-Algorithm
which	O
then	O
must	O
satisfy	O
from	O
which	O
an	O
explicit	O
formula	O
for	O
S−1	O
may	O
be	O
obtained.The	O
set	O
of	O
invertible	O
distributions	O
forms	O
an	O
abelian	O
group	O
under	O
the	O
convolution	B-Algorithm
.	O
</s>
<s>
If	O
f	O
and	O
g	O
are	O
integrable	O
functions	O
,	O
then	O
the	O
integral	O
of	O
their	O
convolution	B-Algorithm
on	O
the	O
whole	O
space	O
is	O
simply	O
obtained	O
as	O
the	O
product	O
of	O
their	O
integrals	O
:	O
</s>
<s>
where	O
d/dx	O
is	O
the	O
derivative	B-Algorithm
.	O
</s>
<s>
More	O
generally	O
,	O
in	O
the	O
case	O
of	O
functions	O
of	O
several	O
variables	O
,	O
an	O
analogous	O
formula	O
holds	O
with	O
the	O
partial	O
derivative	B-Algorithm
:	O
</s>
<s>
A	O
particular	O
consequence	O
of	O
this	O
is	O
that	O
the	O
convolution	B-Algorithm
can	O
be	O
viewed	O
as	O
a	O
"	O
smoothing	O
"	O
operation	O
:	O
the	O
convolution	B-Algorithm
of	O
f	O
and	O
g	O
is	O
differentiable	O
as	O
many	O
times	O
as	O
f	O
and	O
g	O
are	O
in	O
total	O
.	O
</s>
<s>
These	O
identities	O
hold	O
under	O
the	O
precise	O
condition	O
that	O
f	O
and	O
g	O
are	O
absolutely	O
integrable	O
and	O
at	O
least	O
one	O
of	O
them	O
has	O
an	O
absolutely	O
integrable	O
(	O
L1	O
)	O
weak	O
derivative	B-Algorithm
,	O
as	O
a	O
consequence	O
of	O
Young	O
's	O
convolution	B-Algorithm
inequality	O
.	O
</s>
<s>
compactly	O
supported	O
tempered	O
distribution	O
or	O
a	O
Schwartz	B-Algorithm
function	I-Algorithm
and	O
the	O
other	O
is	O
a	O
tempered	O
distribution	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
two	O
positive	O
integrable	O
and	O
infinitely	O
differentiable	O
functions	O
may	O
have	O
a	O
nowhere	O
continuous	O
convolution	B-Algorithm
.	O
</s>
<s>
where	O
denotes	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
,	O
and	O
is	O
a	O
constant	O
that	O
depends	O
on	O
the	O
specific	O
normalization	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
Versions	O
of	O
this	O
theorem	O
also	O
hold	O
for	O
the	O
Laplace	O
transform	O
,	O
two-sided	B-Algorithm
Laplace	I-Algorithm
transform	I-Algorithm
,	O
Z-transform	B-Algorithm
and	O
Mellin	O
transform	O
.	O
</s>
<s>
where	O
is	O
face-splitting	O
product	O
,	O
denotes	O
Kronecker	O
product	O
,	O
denotes	O
Hadamard	O
product	O
(	O
this	O
result	O
is	O
an	O
evolving	O
of	O
count	B-Algorithm
sketch	I-Algorithm
properties	O
)	O
.	O
</s>
<s>
If	O
f	O
is	O
a	O
Schwartz	B-Algorithm
function	I-Algorithm
,	O
then	O
τxf	O
is	O
the	O
convolution	B-Algorithm
with	O
a	O
translated	O
Dirac	O
delta	O
function	O
τxf	O
=	O
f	O
∗	O
τx	O
δ	O
.	O
</s>
<s>
So	O
translation	O
invariance	O
of	O
the	O
convolution	B-Algorithm
of	O
Schwartz	B-Algorithm
functions	I-Algorithm
is	O
a	O
consequence	O
of	O
the	O
associativity	O
of	O
convolution	B-Algorithm
.	O
</s>
<s>
Furthermore	O
,	O
under	O
certain	O
conditions	O
,	O
convolution	B-Algorithm
is	O
the	O
most	O
general	O
translation	O
invariant	O
operation	O
.	O
</s>
<s>
Suppose	O
that	O
S	O
is	O
a	O
bounded	O
linear	B-Architecture
operator	I-Architecture
acting	O
on	O
functions	O
which	O
commutes	O
with	O
translations	O
:	O
S(τxf )	O
=	O
τx(Sf )	O
for	O
all	O
x	O
.	O
</s>
<s>
Then	O
S	O
is	O
given	O
as	O
convolution	B-Algorithm
with	O
a	O
function	O
(	O
or	O
distribution	O
)	O
gS	O
;	O
that	O
is	O
Sf	O
=	O
gS	O
∗	O
f	O
.	O
</s>
<s>
Thus	O
some	O
translation	O
invariant	O
operations	O
can	O
be	O
represented	O
as	O
convolution	B-Algorithm
.	O
</s>
<s>
Convolutions	B-Algorithm
play	O
an	O
important	O
role	O
in	O
the	O
study	O
of	O
time-invariant	B-Algorithm
systems	I-Algorithm
,	O
and	O
especially	O
LTI	O
system	O
theory	O
.	O
</s>
<s>
A	O
more	O
precise	O
version	O
of	O
the	O
theorem	O
quoted	O
above	O
requires	O
specifying	O
the	O
class	O
of	O
functions	O
on	O
which	O
the	O
convolution	B-Algorithm
is	O
defined	O
,	O
and	O
also	O
requires	O
assuming	O
in	O
addition	O
that	O
S	O
must	O
be	O
a	O
continuous	B-Algorithm
linear	I-Algorithm
operator	I-Algorithm
with	O
respect	O
to	O
the	O
appropriate	O
topology	B-Architecture
.	O
</s>
<s>
It	O
is	O
known	O
,	O
for	O
instance	O
,	O
that	O
every	O
continuous	O
translation	O
invariant	O
continuous	B-Algorithm
linear	I-Algorithm
operator	I-Algorithm
on	O
L1	O
is	O
the	O
convolution	B-Algorithm
with	O
a	O
finite	O
Borel	O
measure	O
.	O
</s>
<s>
More	O
generally	O
,	O
every	O
continuous	O
translation	O
invariant	O
continuous	B-Algorithm
linear	I-Algorithm
operator	I-Algorithm
on	O
Lp	O
for	O
1	O
≤	O
p	O
<	O
∞	O
is	O
the	O
convolution	B-Algorithm
with	O
a	O
tempered	O
distribution	O
whose	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
bounded	O
.	O
</s>
<s>
In	O
that	O
case	O
,	O
unless	O
G	O
is	O
unimodular	O
,	O
the	O
convolution	B-Algorithm
defined	O
in	O
this	O
way	O
is	O
not	O
the	O
same	O
as	O
.	O
</s>
<s>
The	O
preference	O
of	O
one	O
over	O
the	O
other	O
is	O
made	O
so	O
that	O
convolution	B-Algorithm
with	O
a	O
fixed	O
function	O
g	O
commutes	O
with	O
left	O
translation	O
in	O
the	O
group	O
:	O
</s>
<s>
Furthermore	O
,	O
the	O
convention	O
is	O
also	O
required	O
for	O
consistency	O
with	O
the	O
definition	O
of	O
the	O
convolution	B-Algorithm
of	O
measures	O
given	O
below	O
.	O
</s>
<s>
On	O
locally	O
compact	O
abelian	O
groups	O
,	O
a	O
version	O
of	O
the	O
convolution	B-Algorithm
theorem	O
holds	O
:	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
a	O
convolution	B-Algorithm
is	O
the	O
pointwise	O
product	O
of	O
the	O
Fourier	B-Algorithm
transforms	I-Algorithm
.	O
</s>
<s>
Consider	O
the	O
family	O
S	O
of	O
operators	O
consisting	O
of	O
all	O
such	O
convolutions	B-Algorithm
and	O
the	O
translation	O
operators	O
.	O
</s>
<s>
According	O
to	O
spectral	O
theory	O
,	O
there	O
exists	O
an	O
orthonormal	O
basis	O
 { hk } 	O
that	O
simultaneously	O
diagonalizes	O
S	O
.	O
This	O
characterizes	O
convolutions	B-Algorithm
on	O
the	O
circle	O
.	O
</s>
<s>
which	O
are	O
precisely	O
the	O
characters	O
of	O
T	O
.	O
Each	O
convolution	B-Algorithm
is	O
a	O
compact	O
multiplication	B-Algorithm
operator	I-Algorithm
in	O
this	O
basis	O
.	O
</s>
<s>
This	O
can	O
be	O
viewed	O
as	O
a	O
version	O
of	O
the	O
convolution	B-Algorithm
theorem	O
discussed	O
above	O
.	O
</s>
<s>
A	O
discrete	O
example	O
is	O
a	O
finite	O
cyclic	O
group	O
of	O
order	O
n	O
.	O
Convolution	B-Algorithm
operators	I-Algorithm
are	O
here	O
represented	O
by	O
circulant	O
matrices	O
,	O
and	O
can	O
be	O
diagonalized	O
by	O
the	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
A	O
similar	O
result	O
holds	O
for	O
compact	O
groups	O
(	O
not	O
necessarily	O
abelian	O
)	O
:	O
the	O
matrix	O
coefficients	O
of	O
finite-dimensional	O
unitary	O
representations	O
form	O
an	O
orthonormal	O
basis	O
in	O
L2	O
by	O
the	O
Peter	O
–	O
Weyl	O
theorem	O
,	O
and	O
an	O
analog	O
of	O
the	O
convolution	B-Algorithm
theorem	O
continues	O
to	O
hold	O
,	O
along	O
with	O
many	O
other	O
aspects	O
of	O
harmonic	O
analysis	O
that	O
depend	O
on	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
In	O
the	O
case	O
when	O
G	O
is	O
locally	O
compact	O
with	O
(	O
left	O
-	O
)	O
Haar	O
measure	O
λ	O
,	O
and	O
μ	O
and	O
ν	O
are	O
absolutely	O
continuous	O
with	O
respect	O
to	O
a	O
λ	O
,	O
so	O
that	O
each	O
has	O
a	O
density	O
function	O
,	O
then	O
the	O
convolution	B-Algorithm
μ∗ν	O
is	O
also	O
absolutely	O
continuous	O
,	O
and	O
its	O
density	O
function	O
is	O
just	O
the	O
convolution	B-Algorithm
of	O
the	O
two	O
separate	O
density	O
functions	O
.	O
</s>
<s>
If	O
μ	O
and	O
ν	O
are	O
probability	O
measures	O
on	O
the	O
topological	O
group	O
then	O
the	O
convolution	B-Algorithm
μ∗ν	O
is	O
the	O
probability	O
distribution	O
of	O
the	O
sum	O
X	O
+	O
Y	O
of	O
two	O
independent	O
random	O
variables	O
X	O
and	O
Y	O
whose	O
respective	O
distributions	O
are	O
μ	O
and	O
ν	O
.	O
</s>
<s>
In	O
convex	O
analysis	O
,	O
the	O
infimal	O
convolution	B-Algorithm
of	O
proper	O
(	O
not	O
identically	O
)	O
convex	O
functions	O
on	O
is	O
defined	O
by	O
:	O
</s>
<s>
It	O
can	O
be	O
shown	O
that	O
the	O
infimal	O
convolution	B-Algorithm
of	O
convex	O
functions	O
is	O
convex	O
.	O
</s>
<s>
Furthermore	O
,	O
it	O
satisfies	O
an	O
identity	O
analogous	O
to	O
that	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
a	O
traditional	O
convolution	B-Algorithm
,	O
with	O
the	O
role	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
played	O
instead	O
by	O
the	O
Legendre	O
transform	O
:	O
</s>
<s>
The	O
convolution	B-Algorithm
is	O
a	O
product	O
defined	O
on	O
the	O
endomorphism	O
algebra	O
End(X )	O
as	O
follows	O
.	O
</s>
<s>
The	O
convolution	B-Algorithm
appears	O
notably	O
in	O
the	O
definition	O
of	O
Hopf	O
algebras	O
.	O
</s>
<s>
Convolution	B-Algorithm
and	O
related	O
operations	O
are	O
found	O
in	O
many	O
applications	O
in	O
science	O
,	O
engineering	O
and	O
mathematics	O
.	O
</s>
<s>
In	O
optics	B-Algorithm
,	O
an	O
out-of-focus	O
photograph	O
is	O
a	O
convolution	B-Algorithm
of	O
the	O
sharp	O
image	O
with	O
a	O
lens	O
function	O
.	O
</s>
<s>
In	O
image	B-Algorithm
processing	I-Algorithm
applications	O
such	O
as	O
adding	O
blurring	O
.	O
</s>
<s>
In	O
statistics	O
,	O
a	O
weighted	O
moving	O
average	O
is	O
a	O
convolution	B-Algorithm
.	O
</s>
<s>
In	O
acoustics	O
,	O
reverberation	O
is	O
the	O
convolution	B-Algorithm
of	O
the	O
original	O
sound	O
with	O
echoes	O
from	O
objects	O
surrounding	O
the	O
sound	O
source	O
.	O
</s>
<s>
In	O
digital	B-General_Concept
signal	I-General_Concept
processing	I-General_Concept
,	O
convolution	B-Algorithm
is	O
used	O
to	O
map	O
the	O
impulse	O
response	O
of	O
a	O
real	O
room	O
on	O
a	O
digital	O
audio	O
signal	O
.	O
</s>
<s>
In	O
electronic	O
music	O
convolution	B-Algorithm
is	O
the	O
imposition	O
of	O
a	O
spectral	O
or	O
rhythmic	O
structure	O
on	O
a	O
sound	O
.	O
</s>
<s>
The	O
convolution	B-Algorithm
of	O
two	O
signals	O
is	O
the	O
filtering	O
of	O
one	O
through	O
the	O
other	O
.	O
</s>
<s>
In	O
electrical	O
engineering	O
,	O
the	O
convolution	B-Algorithm
of	O
one	O
function	O
(	O
the	O
input	O
signal	O
)	O
with	O
a	O
second	O
function	O
(	O
the	O
impulse	O
response	O
)	O
gives	O
the	O
output	O
of	O
a	O
linear	O
time-invariant	B-Algorithm
system	I-Algorithm
(	O
LTI	O
)	O
.	O
</s>
<s>
In	O
physics	O
,	O
wherever	O
there	O
is	O
a	O
linear	O
system	O
with	O
a	O
"	O
superposition	O
principle	O
"	O
,	O
a	O
convolution	B-Algorithm
operation	I-Algorithm
makes	O
an	O
appearance	O
.	O
</s>
<s>
When	O
both	O
effects	O
are	O
operative	O
,	O
the	O
line	O
shape	O
is	O
a	O
convolution	B-Algorithm
of	O
Gaussian	O
and	O
Lorentzian	O
,	O
a	O
Voigt	O
function	O
.	O
</s>
<s>
In	O
computational	O
fluid	O
dynamics	O
,	O
the	O
large	O
eddy	O
simulation	O
(	O
LES	O
)	O
turbulence	O
model	O
uses	O
the	O
convolution	B-Algorithm
operation	I-Algorithm
to	O
lower	O
the	O
range	O
of	O
length	O
scales	O
necessary	O
in	O
computation	O
thereby	O
reducing	O
computational	O
cost	O
.	O
</s>
<s>
In	O
probability	O
theory	O
,	O
the	O
probability	O
distribution	O
of	O
the	O
sum	O
of	O
two	O
independent	O
random	O
variables	O
is	O
the	O
convolution	B-Algorithm
of	O
their	O
individual	O
distributions	O
.	O
</s>
<s>
In	O
kernel	B-General_Concept
density	I-General_Concept
estimation	I-General_Concept
,	O
a	O
distribution	O
is	O
estimated	O
from	O
sample	O
points	O
by	O
convolution	B-Algorithm
with	O
a	O
kernel	B-Algorithm
,	O
such	O
as	O
an	O
isotropic	O
Gaussian	O
.	O
</s>
<s>
In	O
radiotherapy	O
treatment	O
planning	O
systems	O
,	O
most	O
part	O
of	O
all	O
modern	O
codes	O
of	O
calculation	O
applies	O
a	O
convolution-superposition	O
algorithm	O
.	O
</s>
<s>
In	O
structural	O
reliability	O
,	O
the	O
reliability	O
index	O
can	O
be	O
defined	O
based	O
on	O
the	O
convolution	B-Algorithm
theorem	O
.	O
</s>
<s>
In	O
fact	O
,	O
the	O
joint	O
distribution	O
function	O
can	O
be	O
obtained	O
using	O
the	O
convolution	B-Algorithm
theory	O
.	O
</s>
<s>
Convolutional	B-Architecture
neural	I-Architecture
networks	I-Architecture
apply	O
multiple	O
cascaded	O
convolution	B-Algorithm
kernels	B-Algorithm
with	O
applications	O
in	O
machine	B-General_Concept
vision	I-General_Concept
and	O
artificial	B-Application
intelligence	I-Application
.	I-Application
</s>
<s>
Though	O
these	O
are	O
actually	O
cross-correlations	O
rather	O
than	O
convolutions	B-Algorithm
in	O
most	O
cases	O
.	O
</s>
<s>
In	O
Smoothed-particle	B-Application
hydrodynamics	I-Application
,	O
simulations	O
of	O
fluid	O
dynamics	O
are	O
calculated	O
using	O
particles	O
,	O
each	O
with	O
surrounding	O
kernels	B-Algorithm
.	O
</s>
<s>
For	O
any	O
given	O
particle	O
,	O
some	O
physical	O
quantity	O
is	O
calculated	O
as	O
a	O
convolution	B-Algorithm
of	O
with	O
a	O
weighting	O
function	O
,	O
where	O
denotes	O
the	O
neighbors	O
of	O
particle	O
:	O
those	O
that	O
are	O
located	O
within	O
its	O
kernel	B-Algorithm
.	O
</s>
<s>
The	O
convolution	B-Algorithm
is	O
approximated	O
as	O
a	O
summation	O
over	O
each	O
neighbor	O
.	O
</s>
<s>
In	O
Fractional	O
calculus	O
convolution	B-Algorithm
is	O
instrumental	O
in	O
various	O
definitions	O
of	O
fractional	O
integral	O
and	O
fractional	O
derivative	B-Algorithm
.	O
</s>
