<s>
In	O
mathematics	O
,	O
an	O
integral	B-Algorithm
transform	I-Algorithm
maps	O
a	O
function	O
from	O
its	O
original	O
function	B-Algorithm
space	I-Algorithm
into	O
another	O
function	B-Algorithm
space	I-Algorithm
via	O
integration	O
,	O
where	O
some	O
of	O
the	O
properties	O
of	O
the	O
original	O
function	O
might	O
be	O
more	O
easily	O
characterized	O
and	O
manipulated	O
than	O
in	O
the	O
original	O
function	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
The	O
transformed	O
function	O
can	O
generally	O
be	O
mapped	O
back	O
to	O
the	O
original	O
function	B-Algorithm
space	I-Algorithm
using	O
the	O
inverse	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
An	O
integral	B-Algorithm
transform	I-Algorithm
is	O
any	O
transform	B-Algorithm
of	O
the	O
following	O
form	O
:	O
</s>
<s>
The	O
input	O
of	O
this	O
transform	B-Algorithm
is	O
a	O
function	O
,	O
and	O
the	O
output	O
is	O
another	O
function	O
.	O
</s>
<s>
An	O
integral	B-Algorithm
transform	I-Algorithm
is	O
a	O
particular	O
kind	O
of	O
mathematical	O
operator	O
.	O
</s>
<s>
There	O
are	O
numerous	O
useful	O
integral	B-Algorithm
transforms	I-Algorithm
.	O
</s>
<s>
Each	O
is	O
specified	O
by	O
a	O
choice	O
of	O
the	O
function	O
of	O
two	O
variables	O
,	O
the	O
kernel	B-Algorithm
function	O
,	O
integral	B-Algorithm
kernel	I-Algorithm
or	O
nucleus	O
of	O
the	O
transform	B-Algorithm
.	O
</s>
<s>
Some	O
kernels	B-Algorithm
have	O
an	O
associated	O
inverse	B-Algorithm
kernel	I-Algorithm
which	O
(	O
roughly	O
speaking	O
)	O
yields	O
an	O
inverse	B-Algorithm
transform	I-Algorithm
:	O
</s>
<s>
A	O
symmetric	B-Algorithm
kernel	I-Algorithm
is	O
one	O
that	O
is	O
unchanged	O
when	O
the	O
two	O
variables	O
are	O
permuted	O
;	O
it	O
is	O
a	O
kernel	B-Algorithm
function	O
such	O
that	O
.	O
</s>
<s>
In	O
the	O
theory	O
of	O
integral	B-Algorithm
equations	I-Algorithm
,	O
symmetric	B-Algorithm
kernels	I-Algorithm
correspond	O
to	O
self-adjoint	O
operators	O
.	O
</s>
<s>
An	O
integral	B-Algorithm
transform	I-Algorithm
"	O
maps	O
"	O
an	O
equation	O
from	O
its	O
original	O
"	O
domain	O
"	O
into	O
another	O
domain	O
,	O
in	O
which	O
manipulating	O
and	O
solving	O
the	O
equation	O
may	O
be	O
much	O
easier	O
than	O
in	O
the	O
original	O
domain	O
.	O
</s>
<s>
The	O
solution	O
can	O
then	O
be	O
mapped	O
back	O
to	O
the	O
original	O
domain	O
with	O
the	O
inverse	O
of	O
the	O
integral	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
There	O
are	O
many	O
applications	O
of	O
probability	O
that	O
rely	O
on	O
integral	B-Algorithm
transforms	I-Algorithm
,	O
such	O
as	O
"	O
pricing	O
kernel	B-Algorithm
"	O
or	O
stochastic	O
discount	O
factor	O
,	O
or	O
the	O
smoothing	O
of	O
data	O
recovered	O
from	O
robust	O
statistics	O
;	O
see	O
kernel	B-Algorithm
(	O
statistics	O
)	O
.	O
</s>
<s>
Later	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
was	O
developed	O
to	O
remove	O
the	O
requirement	O
of	O
finite	O
intervals	O
.	O
</s>
<s>
As	O
an	O
example	O
of	O
an	O
application	O
of	O
integral	B-Algorithm
transforms	I-Algorithm
,	O
consider	O
the	O
Laplace	O
transform	B-Algorithm
.	O
</s>
<s>
Employing	O
the	O
inverse	B-Algorithm
transform	I-Algorithm
,	O
i.e.	O
,	O
the	O
inverse	O
procedure	O
of	O
the	O
original	O
Laplace	O
transform	B-Algorithm
,	O
one	O
obtains	O
a	O
time-domain	O
solution	O
.	O
</s>
<s>
The	O
Laplace	O
transform	B-Algorithm
finds	O
wide	O
application	O
in	O
physics	O
and	O
particularly	O
in	O
electrical	O
engineering	O
,	O
where	O
the	O
characteristic	O
equations	O
that	O
describe	O
the	O
behavior	O
of	O
an	O
electric	O
circuit	O
in	O
the	O
complex	O
frequency	O
domain	O
correspond	O
to	O
linear	O
combinations	O
of	O
exponentially	O
scaled	O
and	O
time-shifted	O
damped	O
sinusoids	O
in	O
the	O
time	O
domain	O
.	O
</s>
<s>
Other	O
integral	B-Algorithm
transforms	I-Algorithm
find	O
special	O
applicability	O
within	O
other	O
scientific	O
and	O
mathematical	O
disciplines	O
.	O
</s>
<s>
Another	O
usage	O
example	O
is	O
the	O
kernel	B-Algorithm
in	O
the	O
path	O
integral	O
:	O
</s>
<s>
This	O
(	O
physics	O
)	O
kernel	B-Algorithm
is	O
the	O
kernel	B-Algorithm
of	O
the	O
integral	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
However	O
,	O
for	O
each	O
quantum	O
system	O
,	O
there	O
is	O
a	O
different	O
kernel	B-Algorithm
.	O
</s>
<s>
+	O
Table	O
of	O
integral	B-Algorithm
transforms	I-Algorithm
Transform	B-Algorithm
Symbol	O
K	O
f(t )	O
t1	O
t2	O
K−1	O
u1	O
u2	O
Abel	B-Algorithm
transform	I-Algorithm
F	O
,	O
f	O
Assuming	O
the	O
Abel	B-Algorithm
transform	I-Algorithm
is	O
not	O
discontinuous	O
at	O
.	O
</s>
<s>
t	O
Associated	O
Legendre	O
transform	B-Algorithm
Fourier	B-Algorithm
transform	I-Algorithm
Fourier	O
sine	O
transform	B-Algorithm
on	O
,	O
real-valued	O
Fourier	O
cosine	O
transform	B-Algorithm
on	O
,	O
real-valued	O
Hankel	B-Algorithm
transform	I-Algorithm
Hartley	B-Algorithm
transform	I-Algorithm
Hermite	B-Algorithm
transform	I-Algorithm
Hilbert	B-Algorithm
transform	I-Algorithm
Jacobi	O
transform	B-Algorithm
Laguerre	B-Algorithm
transform	I-Algorithm
Laplace	O
transform	B-Algorithm
Legendre	O
transform	B-Algorithm
Mellin	O
transform	B-Algorithm
Some	O
conditions	O
apply	O
,	O
see	O
Mellin	O
inversion	O
theorem	O
for	O
details	O
.	O
</s>
<s>
In	O
the	O
limits	O
of	O
integration	O
for	O
the	O
inverse	B-Algorithm
transform	I-Algorithm
,	O
c	O
is	O
a	O
constant	O
which	O
depends	O
on	O
the	O
nature	O
of	O
the	O
transform	B-Algorithm
function	O
.	O
</s>
<s>
For	O
example	O
,	O
for	O
the	O
one	O
and	O
two-sided	B-Algorithm
Laplace	I-Algorithm
transform	I-Algorithm
,	O
c	O
must	O
be	O
greater	O
than	O
the	O
largest	O
real	O
part	O
of	O
the	O
zeroes	O
of	O
the	O
transform	B-Algorithm
function	O
.	O
</s>
<s>
Note	O
that	O
there	O
are	O
alternative	O
notations	O
and	O
conventions	O
for	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
Here	O
integral	B-Algorithm
transforms	I-Algorithm
are	O
defined	O
for	O
functions	O
on	O
the	O
real	O
numbers	O
,	O
but	O
they	O
can	O
be	O
defined	O
more	O
generally	O
for	O
functions	O
on	O
a	O
group	O
.	O
</s>
<s>
If	O
instead	O
one	O
uses	O
functions	O
on	O
the	O
circle	O
(	O
periodic	O
functions	O
)	O
,	O
integration	O
kernels	B-Algorithm
are	O
then	O
biperiodic	O
functions	O
;	O
convolution	O
by	O
functions	O
on	O
the	O
circle	O
yields	O
circular	B-Algorithm
convolution	I-Algorithm
.	O
</s>
<s>
If	O
one	O
uses	O
functions	O
on	O
the	O
cyclic	O
group	O
of	O
order	O
n	O
(	O
or	O
)	O
,	O
one	O
obtains	O
n	O
×	O
n	O
matrices	O
as	O
integration	O
kernels	B-Algorithm
;	O
convolution	O
corresponds	O
to	O
circulant	O
matrices	O
.	O
</s>
<s>
Although	O
the	O
properties	O
of	O
integral	B-Algorithm
transforms	I-Algorithm
vary	O
widely	O
,	O
they	O
have	O
some	O
properties	O
in	O
common	O
.	O
</s>
<s>
For	O
example	O
,	O
every	O
integral	B-Algorithm
transform	I-Algorithm
is	O
a	O
linear	B-Architecture
operator	I-Architecture
,	O
since	O
the	O
integral	O
is	O
a	O
linear	B-Architecture
operator	I-Architecture
,	O
and	O
in	O
fact	O
if	O
the	O
kernel	B-Algorithm
is	O
allowed	O
to	O
be	O
a	O
generalized	O
function	O
then	O
all	O
linear	O
operators	O
are	O
integral	B-Algorithm
transforms	I-Algorithm
(	O
a	O
properly	O
formulated	O
version	O
of	O
this	O
statement	O
is	O
the	O
Schwartz	O
kernel	B-Algorithm
theorem	O
)	O
.	O
</s>
<s>
The	O
general	O
theory	O
of	O
such	O
integral	B-Algorithm
equations	I-Algorithm
is	O
known	O
as	O
Fredholm	O
theory	O
.	O
</s>
<s>
In	O
this	O
theory	O
,	O
the	O
kernel	B-Algorithm
is	O
understood	O
to	O
be	O
a	O
compact	O
operator	O
acting	O
on	O
a	O
Banach	O
space	B-Algorithm
of	I-Algorithm
functions	I-Algorithm
.	O
</s>
<s>
Depending	O
on	O
the	O
situation	O
,	O
the	O
kernel	B-Algorithm
is	O
then	O
variously	O
referred	O
to	O
as	O
the	O
Fredholm	O
operator	O
,	O
the	O
nuclear	B-Algorithm
operator	I-Algorithm
or	O
the	O
Fredholm	O
kernel	B-Algorithm
.	O
</s>
