<s>
In	O
mathematics	O
,	O
the	O
Fredholm	B-Algorithm
integral	I-Algorithm
equation	I-Algorithm
is	O
an	O
integral	B-Algorithm
equation	I-Algorithm
whose	O
solution	O
gives	O
rise	O
to	O
Fredholm	O
theory	O
,	O
the	O
study	O
of	O
Fredholm	O
kernels	B-Algorithm
and	O
Fredholm	O
operators	O
.	O
</s>
<s>
The	O
integral	B-Algorithm
equation	I-Algorithm
was	O
studied	O
by	O
Ivar	O
Fredholm	O
.	O
</s>
<s>
A	O
Fredholm	B-Algorithm
equation	I-Algorithm
is	O
an	O
integral	B-Algorithm
equation	I-Algorithm
in	O
which	O
the	O
term	O
containing	O
the	O
kernel	B-Algorithm
function	O
(	O
defined	O
below	O
)	O
has	O
constants	O
as	O
integration	O
limits	O
.	O
</s>
<s>
A	O
closely	O
related	O
form	O
is	O
the	O
Volterra	B-Algorithm
integral	I-Algorithm
equation	I-Algorithm
which	O
has	O
variable	O
integral	O
limits	O
.	O
</s>
<s>
and	O
the	O
problem	O
is	O
,	O
given	O
the	O
continuous	O
kernel	B-Algorithm
function	O
and	O
the	O
function	O
,	O
to	O
find	O
the	O
function	O
.	O
</s>
<s>
where	O
and	O
are	O
the	O
direct	O
and	O
inverse	O
Fourier	B-Algorithm
transforms	I-Algorithm
,	O
respectively	O
.	O
</s>
<s>
This	O
case	O
would	O
not	O
be	O
typically	O
included	O
under	O
the	O
umbrella	O
of	O
Fredholm	B-Algorithm
integral	I-Algorithm
equations	I-Algorithm
,	O
a	O
name	O
that	O
is	O
usually	O
reserved	O
for	O
when	O
the	O
integral	O
operator	O
defines	O
a	O
compact	O
operator	O
(	O
convolution	B-Language
operators	O
on	O
non-compact	O
groups	O
are	O
non-compact	O
,	O
since	O
,	O
in	O
general	O
,	O
the	O
spectrum	O
of	O
the	O
operator	O
of	O
convolution	B-Language
with	O
contains	O
the	O
range	O
of	O
,	O
which	O
is	O
usually	O
a	O
non-countable	O
set	O
,	O
whereas	O
compact	O
operators	O
have	O
discrete	O
countable	O
spectra	O
)	O
.	O
</s>
<s>
Given	O
the	O
kernel	B-Algorithm
,	O
and	O
the	O
function	O
,	O
the	O
problem	O
is	O
typically	O
to	O
find	O
the	O
function	O
.	O
</s>
<s>
The	O
general	O
theory	O
underlying	O
the	O
Fredholm	B-Algorithm
equations	I-Algorithm
is	O
known	O
as	O
Fredholm	O
theory	O
.	O
</s>
<s>
One	O
of	O
the	O
principal	O
results	O
is	O
that	O
the	O
kernel	B-Algorithm
yields	O
a	O
compact	O
operator	O
.	O
</s>
<s>
Fredholm	B-Algorithm
equations	I-Algorithm
arise	O
naturally	O
in	O
the	O
theory	O
of	O
signal	O
processing	O
,	O
for	O
example	O
as	O
the	O
famous	O
spectral	B-Algorithm
concentration	I-Algorithm
problem	I-Algorithm
popularized	O
by	O
David	O
Slepian	O
.	O
</s>
<s>
In	O
physics	O
,	O
the	O
solution	O
of	O
such	O
integral	B-Algorithm
equations	I-Algorithm
allows	O
for	O
experimental	O
spectra	O
to	O
be	O
related	O
to	O
various	O
underlying	O
distributions	O
,	O
for	O
instance	O
the	O
mass	O
distribution	O
of	O
polymers	O
in	O
a	O
polymeric	O
melt	O
,	O
</s>
<s>
In	O
addition	O
,	O
Fredholm	B-Algorithm
integral	I-Algorithm
equations	I-Algorithm
also	O
arise	O
in	O
fluid	O
mechanics	O
problems	O
involving	O
hydrodynamic	O
interactions	O
near	O
finite-sized	O
elastic	O
interfaces	O
.	O
</s>
<s>
A	O
specific	O
application	O
of	O
Fredholm	B-Algorithm
equation	I-Algorithm
is	O
the	O
generation	O
of	O
photo-realistic	O
images	O
in	O
computer	O
graphics	O
,	O
in	O
which	O
the	O
Fredholm	B-Algorithm
equation	I-Algorithm
is	O
used	O
to	O
model	O
light	O
transport	O
from	O
the	O
virtual	O
light	O
sources	O
to	O
the	O
image	O
plane	O
.	O
</s>
<s>
The	O
Fredholm	B-Algorithm
equation	I-Algorithm
is	O
often	O
called	O
the	O
rendering	O
equation	O
in	O
this	O
context	O
.	O
</s>
