<s>
In	O
physics	O
and	O
mathematics	O
,	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
FT	O
)	O
is	O
a	O
transform	B-Algorithm
that	O
converts	O
a	O
function	O
into	O
a	O
form	O
that	O
describes	O
the	O
frequencies	O
present	O
in	O
the	O
original	O
function	O
.	O
</s>
<s>
The	O
output	O
of	O
the	O
transform	B-Algorithm
is	O
a	O
complex-valued	O
function	O
of	O
frequency	O
.	O
</s>
<s>
The	O
term	O
Fourier	B-Algorithm
transform	I-Algorithm
refers	O
to	O
both	O
this	O
complex-valued	O
function	O
and	O
the	O
mathematical	O
operation	O
.	O
</s>
<s>
When	O
a	O
distinction	O
needs	O
to	O
be	O
made	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
sometimes	O
called	O
the	O
frequency	O
domain	B-Algorithm
representation	O
of	O
the	O
original	O
function	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
analogous	O
to	O
decomposing	O
the	O
sound	O
of	O
a	O
musical	O
chord	O
into	O
terms	O
of	O
the	O
intensity	O
of	O
its	O
constituent	O
pitches	O
.	O
</s>
<s>
Functions	O
that	O
are	O
localized	O
in	O
the	O
time	O
domain	B-Algorithm
have	O
Fourier	B-Algorithm
transforms	I-Algorithm
that	O
are	O
spread	O
out	O
across	O
the	O
frequency	O
domain	B-Algorithm
and	O
vice	O
versa	O
,	O
a	O
phenomenon	O
known	O
as	O
the	O
uncertainty	O
principle	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
a	O
Gaussian	O
function	O
is	O
another	O
Gaussian	O
function	O
.	O
</s>
<s>
Joseph	O
Fourier	O
introduced	O
the	O
transform	B-Algorithm
in	O
his	O
study	O
of	O
heat	O
transfer	O
,	O
where	O
Gaussian	O
functions	O
appear	O
as	O
solutions	O
of	O
the	O
heat	O
equation	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
can	O
be	O
formally	O
defined	O
as	O
an	O
improper	O
Riemann	O
integral	O
,	O
making	O
it	O
an	O
integral	B-Algorithm
transform	I-Algorithm
,	O
although	O
this	O
definition	O
is	O
not	O
suitable	O
for	O
many	O
applications	O
requiring	O
a	O
more	O
sophisticated	O
integration	O
theory	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
can	O
also	O
be	O
generalized	O
to	O
functions	O
of	O
several	O
variables	O
on	O
Euclidean	O
space	O
,	O
sending	O
a	O
function	O
of	O
'	O
position	O
space	O
 '	O
to	O
a	O
function	O
of	O
momentum	B-Algorithm
(	O
or	O
a	O
function	O
of	O
space	O
and	O
time	O
to	O
a	O
function	O
of	O
4-momentum	O
)	O
.	O
</s>
<s>
This	O
idea	O
makes	O
the	O
spatial	O
Fourier	B-Algorithm
transform	I-Algorithm
very	O
natural	O
in	O
the	O
study	O
of	O
waves	O
,	O
as	O
well	O
as	O
in	O
quantum	O
mechanics	O
,	O
where	O
it	O
is	O
important	O
to	O
be	O
able	O
to	O
represent	O
wave	O
solutions	O
as	O
functions	O
of	O
either	O
position	O
or	O
momentum	B-Algorithm
and	O
sometimes	O
both	O
.	O
</s>
<s>
Still	O
further	O
generalization	O
is	O
possible	O
to	O
functions	O
on	O
groups	O
,	O
which	O
,	O
besides	O
the	O
original	O
Fourier	B-Algorithm
transform	I-Algorithm
on	O
or	O
(	O
viewed	O
as	O
groups	O
under	O
addition	O
)	O
,	O
notably	O
includes	O
the	O
discrete-time	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
DTFT	O
,	O
group	O
=	O
)	O
,	O
the	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
(	O
DFT	O
,	O
group	O
=	O
)	O
and	O
the	O
Fourier	O
series	O
or	O
circular	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
group	O
=	O
,	O
the	O
unit	O
circle	O
≈	O
closed	O
finite	O
interval	O
with	O
endpoints	O
identified	O
)	O
.	O
</s>
<s>
The	O
fast	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
FFT	O
)	O
is	O
an	O
algorithm	O
for	O
computing	O
the	O
DFT	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
an	O
extension	O
of	O
the	O
Fourier	O
series	O
,	O
which	O
in	O
its	O
most	O
general	O
form	O
introduces	O
the	O
use	O
of	O
complex	O
exponential	O
functions	O
.	O
</s>
<s>
For	O
example	O
,	O
for	O
a	O
function	O
,	O
the	O
amplitude	B-Application
and	O
phase	O
of	O
a	O
frequency	O
component	O
at	O
frequency	O
,	O
is	O
given	O
by	O
this	O
complex	O
number	O
:	O
</s>
<s>
The	O
extension	O
provides	O
a	O
frequency	O
continuum	O
of	O
components	O
using	O
an	O
infinite	O
domain	B-Algorithm
of	O
integration	O
:	O
</s>
<s>
Here	O
,	O
the	O
transform	B-Algorithm
of	O
function	O
at	O
frequency	O
is	O
denoted	O
by	O
the	O
complex	O
number	O
,	O
which	O
is	O
just	O
one	O
of	O
several	O
common	O
conventions	O
.	O
</s>
<s>
Evaluating	O
for	O
all	O
values	O
of	O
produces	O
the	O
frequency-domain	O
function	O
.	O
</s>
<s>
When	O
the	O
independent	O
variable	O
(	O
)	O
represents	O
time	O
(	O
often	O
denoted	O
by	O
)	O
,	O
the	O
transform	B-Algorithm
variable	O
(	O
)	O
represents	O
frequency	O
(	O
often	O
denoted	O
by	O
)	O
.	O
</s>
<s>
For	O
each	O
frequency	O
,	O
the	O
magnitude	O
(	O
absolute	O
value	O
)	O
of	O
the	O
complex	O
value	O
represents	O
the	O
amplitude	B-Application
of	O
a	O
constituent	O
complex	O
sinusoid	O
with	O
that	O
frequency	O
integrated	O
over	O
the	O
domain	B-Algorithm
,	O
and	O
the	O
argument	O
of	O
the	O
complex	O
value	O
represents	O
that	O
complex	O
sinusoid	O
's	O
phase	O
offset	O
.	O
</s>
<s>
If	O
a	O
frequency	O
is	O
not	O
present	O
,	O
the	O
transform	B-Algorithm
has	O
a	O
value	O
of	O
0	O
for	O
that	O
frequency	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
not	O
limited	O
to	O
functions	O
of	O
time	O
,	O
but	O
the	O
domain	B-Algorithm
of	O
the	O
original	O
function	O
is	O
commonly	O
referred	O
to	O
as	O
the	O
time	O
domain	B-Algorithm
.	O
</s>
<s>
The	O
Fourier	O
inversion	O
theorem	O
provides	O
a	O
synthesis	O
process	O
that	O
recreates	O
the	O
original	O
function	O
from	O
its	O
frequency	O
domain	B-Algorithm
representation	O
.	O
</s>
<s>
A	O
key	O
to	O
interpreting	O
is	O
that	O
the	O
effect	O
of	O
multiplying	O
by	O
is	O
to	O
subtract	O
from	O
every	O
frequency	O
component	O
of	O
function	O
(	O
also	O
see	O
Negative	O
frequency	O
)	O
So	O
the	O
component	O
that	O
was	O
at	O
ends	O
up	O
at	O
zero	O
hertz	O
,	O
and	O
the	O
integral	O
produces	O
its	O
amplitude	B-Application
,	O
because	O
all	O
the	O
other	O
components	O
are	O
oscillatory	O
and	O
integrate	O
to	O
zero	O
over	O
an	O
infinite	O
interval	O
.	O
</s>
<s>
The	O
functions	O
and	O
are	O
often	O
referred	O
to	O
as	O
a	O
Fourier	B-Algorithm
transform	I-Algorithm
pair	O
.	O
</s>
<s>
A	O
common	O
notation	O
for	O
designating	O
transform	B-Algorithm
pairs	O
is	O
:	O
</s>
<s>
Similarly	O
,	O
under	O
suitable	O
conditions	O
on	O
,	O
the	O
Fourier	O
inversion	O
formula	O
on	O
is:The	O
complex	O
number	O
,	O
,	O
conveys	O
both	O
amplitude	B-Application
and	O
phase	O
of	O
frequency	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
on	O
Euclidean	O
space	O
is	O
treated	O
separately	O
,	O
in	O
which	O
the	O
variable	O
often	O
represents	O
position	O
and	O
momentum	B-Algorithm
.	O
</s>
<s>
The	O
conventions	O
chosen	O
in	O
this	O
article	O
are	O
those	O
of	O
harmonic	O
analysis	O
,	O
and	O
are	O
characterized	O
as	O
the	O
unique	O
conventions	O
such	O
that	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
both	O
unitary	B-Algorithm
on	O
and	O
an	O
algebra	O
homomorphism	O
from	O
to	O
,	O
without	O
renormalizing	O
the	O
Lebesgue	O
measure	O
.	O
</s>
<s>
Many	O
other	O
characterizations	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
exist	O
.	O
</s>
<s>
For	O
example	O
,	O
one	O
uses	O
the	O
Stone	O
–	O
von	O
Neumann	O
theorem	O
:	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
the	O
unique	O
unitary	B-Algorithm
intertwiner	O
for	O
the	O
symplectic	O
and	O
Euclidean	O
Schrödinger	O
representations	O
of	O
the	O
Heisenberg	O
group	O
.	O
</s>
<s>
That	O
important	O
work	O
was	O
corrected	O
and	O
expanded	O
upon	O
by	O
others	O
to	O
provide	O
the	O
foundation	O
for	O
the	O
various	O
forms	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
used	O
since	O
.	O
</s>
<s>
One	O
aspect	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
that	O
is	O
often	O
confusing	O
is	O
its	O
use	O
of	O
negative	O
frequency	O
.	O
</s>
<s>
For	O
real-valued	O
functions	O
,	O
there	O
is	O
a	O
simple	O
relationship	O
between	O
the	O
values	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
for	O
positive	O
and	O
negative	O
(	O
see	O
conjugation	O
below	O
)	O
.	O
</s>
<s>
One	O
reason	O
for	O
this	O
is	O
that	O
many	O
applications	O
have	O
to	O
take	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
complex-valued	O
functions	O
,	O
such	O
as	O
partial	O
differential	O
equations	O
,	O
radar	B-Application
,	O
nonlinear	O
optics	O
,	O
quantum	O
mechanics	O
,	O
and	O
others	O
.	O
</s>
<s>
In	O
these	O
cases	O
,	O
the	O
value	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
at	O
negative	O
frequencies	O
is	O
distinct	O
from	O
the	O
value	O
at	O
real	O
frequencies	O
,	O
and	O
they	O
are	O
important	O
.	O
</s>
<s>
In	O
these	O
situations	O
,	O
the	O
concept	O
of	O
what	O
is	O
a	O
frequency	O
is	O
defined	O
by	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
rather	O
than	O
appealing	O
to	O
a	O
rate	O
or	O
period	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
a	O
periodic	O
function	O
cannot	O
be	O
defined	O
using	O
the	O
integral	O
formula	O
directly	O
.	O
</s>
<s>
This	O
makes	O
it	O
possible	O
to	O
see	O
a	O
connection	O
between	O
the	O
Fourier	O
series	O
and	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
for	O
periodic	O
functions	O
which	O
have	O
a	O
convergent	O
Fourier	O
series	O
.	O
</s>
<s>
In	O
other	O
words	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
a	O
Dirac	O
comb	O
function	O
whose	O
teeth	O
are	O
multiplied	O
by	O
the	O
Fourier	O
series	O
coefficients	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
an	O
integrable	O
function	O
can	O
be	O
sampled	O
at	O
regular	O
intervals	O
of	O
These	O
samples	O
can	O
be	O
deduced	O
from	O
one	O
cycle	O
of	O
a	O
periodic	O
function	O
which	O
has	O
Fourier	O
series	O
coefficients	O
proportional	O
to	O
those	O
samples	O
by	O
the	O
Poisson	O
summation	O
formula	O
:	O
</s>
<s>
The	O
integrability	O
of	O
ensures	O
the	O
periodic	B-Algorithm
summation	I-Algorithm
converges	O
.	O
</s>
<s>
The	O
following	O
figures	O
provide	O
a	O
visual	O
illustration	O
of	O
how	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
measures	O
whether	O
a	O
frequency	O
is	O
present	O
in	O
a	O
particular	O
function	O
.	O
</s>
<s>
This	O
function	O
was	O
specially	O
chosen	O
to	O
have	O
a	O
real	O
Fourier	B-Algorithm
transform	I-Algorithm
that	O
can	O
be	O
easily	O
plotted	O
.	O
</s>
<s>
Therefore	O
,	O
in	O
this	O
case	O
,	O
the	O
integrand	O
oscillates	O
fast	O
enough	O
so	O
that	O
the	O
integral	O
is	O
very	O
small	O
and	O
the	O
value	O
for	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
for	O
that	O
frequency	O
is	O
nearly	O
zero	O
.	O
</s>
<s>
The	O
general	O
situation	O
may	O
be	O
a	O
bit	O
more	O
complicated	O
than	O
this	O
,	O
but	O
this	O
in	O
spirit	O
is	O
how	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
measures	O
how	O
much	O
of	O
an	O
individual	O
frequency	O
is	O
present	O
in	O
a	O
function	O
.	O
</s>
<s>
We	O
denote	O
the	O
Fourier	B-Algorithm
transforms	I-Algorithm
of	O
these	O
functions	O
as	O
,	O
and	O
respectively	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
has	O
the	O
following	O
basic	O
properties	O
:	O
</s>
<s>
And	O
there	O
is	O
a	O
one-to-one	O
mapping	O
between	O
the	O
four	O
components	O
of	O
a	O
complex	O
time	O
function	O
and	O
the	O
four	O
components	O
of	O
its	O
complex	O
frequency	B-Algorithm
transform	I-Algorithm
:	O
</s>
<s>
The	O
transform	B-Algorithm
of	O
a	O
real-valued	O
function	O
(	O
)	O
is	O
the	O
even	O
symmetric	O
function	O
.	O
</s>
<s>
Conversely	O
,	O
an	O
even-symmetric	O
transform	B-Algorithm
implies	O
a	O
real-valued	O
time-domain	O
.	O
</s>
<s>
The	O
transform	B-Algorithm
of	O
an	O
imaginary-valued	O
function	O
(	O
)	O
is	O
the	O
odd	O
symmetric	O
function	O
,	O
and	O
the	O
converse	O
is	O
true	O
.	O
</s>
<s>
The	O
transform	B-Algorithm
of	O
an	O
even-symmetric	O
function	O
(	O
)	O
is	O
the	O
real-valued	O
function	O
,	O
and	O
the	O
converse	O
is	O
true	O
.	O
</s>
<s>
The	O
transform	B-Algorithm
of	O
an	O
odd-symmetric	O
function	O
(	O
)	O
is	O
the	O
imaginary-valued	O
function	O
,	O
and	O
the	O
converse	O
is	O
true	O
.	O
</s>
<s>
That	O
is	O
the	O
same	O
as	O
the	O
integral	O
of	O
over	O
all	O
its	O
domain	B-Algorithm
and	O
is	O
also	O
known	O
as	O
the	O
average	O
value	O
or	O
DC	O
bias	O
of	O
the	O
function	O
.	O
</s>
<s>
Under	O
suitable	O
conditions	O
on	O
the	O
function	O
,	O
it	O
can	O
be	O
recovered	O
from	O
its	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
Indeed	O
,	O
denoting	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
operator	O
by	O
,	O
so	O
,	O
then	O
for	O
suitable	O
functions	O
,	O
applying	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
twice	O
simply	O
flips	O
the	O
function	O
:	O
,	O
which	O
can	O
be	O
interpreted	O
as	O
"	O
reversing	O
time	O
"	O
.	O
</s>
<s>
Since	O
reversing	O
time	O
is	O
two-periodic	O
,	O
applying	O
this	O
twice	O
yields	O
,	O
so	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
operator	O
is	O
four-periodic	O
,	O
and	O
similarly	O
the	O
inverse	O
Fourier	B-Algorithm
transform	I-Algorithm
can	O
be	O
obtained	O
by	O
applying	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
three	O
times	O
:	O
.	O
</s>
<s>
In	O
particular	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
invertible	O
(	O
under	O
suitable	O
conditions	O
)	O
.	O
</s>
<s>
This	O
fourfold	O
periodicity	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
similar	O
to	O
a	O
rotation	O
of	O
the	O
plane	O
by	O
90°	O
,	O
particularly	O
as	O
the	O
two-fold	O
iteration	O
yields	O
a	O
reversal	O
,	O
and	O
in	O
fact	O
this	O
analogy	O
can	O
be	O
made	O
precise	O
.	O
</s>
<s>
While	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
can	O
simply	O
be	O
interpreted	O
as	O
switching	O
the	O
time	O
domain	B-Algorithm
and	O
the	O
frequency	O
domain	B-Algorithm
,	O
with	O
the	O
inverse	O
Fourier	B-Algorithm
transform	I-Algorithm
switching	O
them	O
back	O
,	O
more	O
geometrically	O
it	O
can	O
be	O
interpreted	O
as	O
a	O
rotation	O
by	O
90°	O
in	O
the	O
time	O
–	O
frequency	O
domain	B-Algorithm
(	O
considering	O
time	O
as	O
the	O
-axis	O
and	O
frequency	O
as	O
the	O
-axis	O
)	O
,	O
and	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
can	O
be	O
generalized	O
to	O
the	O
fractional	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
which	O
involves	O
rotations	O
by	O
other	O
angles	O
.	O
</s>
<s>
This	O
can	O
be	O
further	O
generalized	O
to	O
linear	B-Algorithm
canonical	I-Algorithm
transformations	I-Algorithm
,	O
which	O
can	O
be	O
visualized	O
as	O
the	O
action	O
of	O
the	O
special	O
linear	O
group	O
on	O
the	O
time	O
–	O
frequency	O
plane	O
,	O
with	O
the	O
preserved	O
symplectic	O
form	O
corresponding	O
to	O
the	O
uncertainty	O
principle	O
,	O
below	O
.	O
</s>
<s>
This	O
approach	O
is	O
particularly	O
studied	O
in	O
signal	O
processing	O
,	O
under	O
time	B-Algorithm
–	I-Algorithm
frequency	I-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
The	O
frequency	O
variable	O
must	O
have	O
inverse	O
units	O
to	O
the	O
units	O
of	O
the	O
original	O
function	O
's	O
domain	B-Algorithm
(	O
typically	O
named	O
or	O
)	O
.	O
</s>
<s>
That	O
is	O
to	O
say	O
,	O
there	O
are	O
two	O
versions	O
of	O
the	O
real	O
line	O
:	O
one	O
which	O
is	O
the	O
range	B-Algorithm
of	O
and	O
measured	O
in	O
units	O
of	O
,	O
and	O
the	O
other	O
which	O
is	O
the	O
range	B-Algorithm
of	O
and	O
measured	O
in	O
inverse	O
units	O
to	O
the	O
units	O
of	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
goes	O
from	O
one	O
space	O
of	O
functions	O
to	O
a	O
different	O
space	O
of	O
functions	O
:	O
functions	O
which	O
have	O
a	O
different	O
domain	B-Algorithm
of	O
definition	O
.	O
</s>
<s>
In	O
general	O
,	O
must	O
always	O
be	O
taken	O
to	O
be	O
a	O
linear	B-Algorithm
form	I-Algorithm
on	O
the	O
space	O
of	O
its	O
domain	B-Algorithm
,	O
which	O
is	O
to	O
say	O
that	O
the	O
second	O
real	O
line	O
is	O
the	O
dual	O
space	O
of	O
the	O
first	O
real	O
line	O
.	O
</s>
<s>
See	O
the	O
article	O
on	O
linear	B-Language
algebra	I-Language
for	O
a	O
more	O
formal	O
explanation	O
and	O
for	O
more	O
details	O
.	O
</s>
<s>
This	O
point	O
of	O
view	O
becomes	O
essential	O
in	O
generalisations	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
to	O
general	O
symmetry	O
groups	O
,	O
including	O
the	O
case	O
of	O
Fourier	O
series	O
.	O
</s>
<s>
That	O
there	O
is	O
no	O
one	O
preferred	O
way	O
(	O
often	O
,	O
one	O
says	O
"	O
no	O
canonical	O
way	O
"	O
)	O
to	O
compare	O
the	O
two	O
versions	O
of	O
the	O
real	O
line	O
which	O
are	O
involved	O
in	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
—	O
fixing	O
the	O
units	O
on	O
one	O
line	O
does	O
not	O
force	O
the	O
scale	O
of	O
the	O
units	O
on	O
the	O
other	O
line	O
—	O
is	O
the	O
reason	O
for	O
the	O
plethora	O
of	O
rival	O
conventions	O
on	O
the	O
definition	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
Let	O
be	O
the	O
form	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
in	O
terms	O
of	O
ordinary	O
frequency	O
.	O
</s>
<s>
In	O
other	O
conventions	O
,	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
has	O
in	O
the	O
exponent	O
instead	O
of	O
,	O
and	O
vice	O
versa	O
for	O
the	O
inversion	O
formula	O
.	O
</s>
<s>
It	O
simply	O
means	O
that	O
is	O
the	O
amplitude	B-Application
of	O
the	O
wave	O
instead	O
of	O
the	O
wave	O
(	O
the	O
former	O
,	O
with	O
its	O
minus	O
sign	O
,	O
is	O
often	O
seen	O
in	O
the	O
time	O
dependence	O
for	O
Sinusoidal	O
plane-wave	O
solutions	O
of	O
the	O
electromagnetic	O
wave	O
equation	O
,	O
or	O
in	O
the	O
time	O
dependence	O
for	O
quantum	O
wave	O
functions	O
)	O
.	O
</s>
<s>
Many	O
of	O
the	O
identities	O
involving	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
remain	O
valid	O
in	O
those	O
conventions	O
,	O
provided	O
all	O
terms	O
that	O
explicitly	O
involve	O
have	O
it	O
replaced	O
by	O
.	O
</s>
<s>
When	O
using	O
dimensionless	O
units	O
,	O
the	O
constant	O
factors	O
might	O
not	O
even	O
be	O
written	O
in	O
the	O
transform	B-Algorithm
definition	O
.	O
</s>
<s>
(	O
In	O
probability	O
theory	O
,	O
and	O
in	O
mathematical	O
statistics	O
,	O
the	O
use	O
of	O
the	O
Fourier	O
—	O
Stieltjes	O
transform	B-Algorithm
is	O
preferred	O
,	O
because	O
so	O
many	O
random	O
variables	O
are	O
not	O
of	O
continuous	O
type	O
,	O
and	O
do	O
not	O
possess	O
a	O
density	O
function	O
,	O
and	O
one	O
must	O
treat	O
not	O
functions	O
but	O
distributions	O
,	O
i.e.	O
,	O
measures	O
which	O
possess	O
"	O
atoms	O
"	O
.	O
)	O
</s>
<s>
From	O
the	O
higher	O
point	O
of	O
view	O
of	O
group	O
characters	O
,	O
which	O
is	O
much	O
more	O
abstract	O
,	O
all	O
these	O
arbitrary	O
choices	O
disappear	O
,	O
as	O
will	O
be	O
explained	O
in	O
the	O
later	O
section	O
of	O
this	O
article	O
,	O
which	O
treats	O
the	O
notion	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
a	O
function	O
on	O
a	O
locally	O
compact	O
Abelian	O
group	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
may	O
be	O
defined	O
in	O
some	O
cases	O
for	O
non-integrable	O
functions	O
,	O
but	O
the	O
Fourier	B-Algorithm
transforms	I-Algorithm
of	O
integrable	O
functions	O
have	O
several	O
strong	O
properties	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
the	O
rectangular	O
function	O
,	O
which	O
is	O
integrable	O
,	O
is	O
the	O
sinc	B-Algorithm
function	I-Algorithm
,	O
which	O
is	O
not	O
Lebesgue	O
integrable	O
,	O
because	O
its	O
improper	O
integrals	O
behave	O
analogously	O
to	O
the	O
alternating	O
harmonic	O
series	O
,	O
in	O
converging	O
to	O
a	O
sum	O
without	O
being	O
absolutely	O
convergent	O
.	O
</s>
<s>
It	O
is	O
not	O
generally	O
possible	O
to	O
write	O
the	O
inverse	O
transform	B-Algorithm
as	O
a	O
Lebesgue	O
integral	O
.	O
</s>
<s>
That	O
is	O
,	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
injective	O
on	O
.	O
</s>
<s>
Let	O
and	O
be	O
integrable	O
,	O
and	O
let	O
and	O
be	O
their	O
Fourier	B-Algorithm
transforms	I-Algorithm
.	O
</s>
<s>
If	O
and	O
are	O
also	O
square-integrable	B-Algorithm
,	O
then	O
the	O
Parseval	O
formula	O
follows	O
:	O
</s>
<s>
Plancherel	O
's	O
theorem	O
makes	O
it	O
possible	O
to	O
extend	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
by	O
a	O
continuity	O
argument	O
,	O
to	O
a	O
unitary	B-Algorithm
operator	I-Algorithm
on	O
.	O
</s>
<s>
On	O
,	O
this	O
extension	O
agrees	O
with	O
original	O
Fourier	B-Algorithm
transform	I-Algorithm
defined	O
on	O
,	O
thus	O
enlarging	O
the	O
domain	B-Algorithm
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
to	O
(	O
and	O
consequently	O
to	O
for	O
)	O
.	O
</s>
<s>
Plancherel	O
's	O
theorem	O
has	O
the	O
interpretation	O
in	O
the	O
sciences	O
that	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
preserves	O
the	O
energy	O
of	O
the	O
original	O
quantity	O
.	O
</s>
<s>
But	O
Parseval	O
's	O
formula	O
makes	O
sense	O
for	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
as	O
well	O
,	O
and	O
so	O
even	O
though	O
in	O
the	O
context	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
it	O
was	O
proved	O
by	O
Plancherel	O
,	O
it	O
is	O
still	O
often	O
referred	O
to	O
as	O
Parseval	O
's	O
formula	O
,	O
or	O
Parseval	O
's	O
relation	O
,	O
or	O
even	O
Parseval	O
's	O
theorem	O
.	O
</s>
<s>
The	O
Poisson	O
summation	O
formula	O
(	O
PSF	O
)	O
is	O
an	O
equation	O
that	O
relates	O
the	O
Fourier	O
series	O
coefficients	O
of	O
the	O
periodic	B-Algorithm
summation	I-Algorithm
of	O
a	O
function	O
to	O
values	O
of	O
the	O
function	O
's	O
continuous	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
It	O
has	O
a	O
variety	O
of	O
useful	O
forms	O
that	O
are	O
derived	O
from	O
the	O
basic	O
one	O
by	O
application	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
's	O
scaling	O
and	O
time-shifting	O
properties	O
.	O
</s>
<s>
The	O
frequency-domain	O
dual	O
of	O
the	O
standard	O
Poisson	O
summation	O
formula	O
is	O
also	O
called	O
the	O
discrete-time	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
Suppose	O
is	O
an	O
absolutely	O
continuous	O
differentiable	O
function	O
,	O
and	O
both	O
and	O
its	O
derivative	B-Algorithm
are	O
integrable	O
.	O
</s>
<s>
By	O
applying	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
and	O
using	O
these	O
formulas	O
,	O
some	O
ordinary	O
differential	O
equations	O
can	O
be	O
transformed	O
into	O
algebraic	O
equations	O
,	O
which	O
are	O
much	O
easier	O
to	O
solve	O
.	O
</s>
<s>
By	O
using	O
the	O
analogous	O
rules	O
for	O
the	O
inverse	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
one	O
can	O
also	O
say	O
"	O
quickly	O
falls	O
to	O
0	O
for	O
if	O
and	O
only	O
if	O
is	O
smooth.	O
"	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
translates	B-Algorithm
between	O
convolution	B-Language
and	O
multiplication	O
of	O
functions	O
.	O
</s>
<s>
If	O
and	O
are	O
integrable	O
functions	O
with	O
Fourier	B-Algorithm
transforms	I-Algorithm
and	O
respectively	O
,	O
then	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
the	O
convolution	B-Language
is	O
given	O
by	O
the	O
product	O
of	O
the	O
Fourier	B-Algorithm
transforms	I-Algorithm
and	O
(	O
under	O
other	O
conventions	O
for	O
the	O
definition	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
a	O
constant	O
factor	O
may	O
appear	O
)	O
.	O
</s>
<s>
where	O
denotes	O
the	O
convolution	B-Language
operation	O
,	O
then	O
:	O
</s>
<s>
Conversely	O
,	O
if	O
can	O
be	O
decomposed	O
as	O
the	O
product	O
of	O
two	O
square	B-Algorithm
integrable	I-Algorithm
functions	I-Algorithm
and	O
,	O
then	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
is	O
given	O
by	O
the	O
convolution	B-Language
of	O
the	O
respective	O
Fourier	B-Algorithm
transforms	I-Algorithm
and	O
.	O
</s>
<s>
then	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
is	O
:	O
</s>
<s>
leads	O
to	O
eigenfunctions	B-Algorithm
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
as	O
long	O
as	O
the	O
form	O
of	O
the	O
equation	O
remains	O
invariant	O
under	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
In	O
other	O
words	O
,	O
every	O
solution	O
and	O
its	O
Fourier	B-Algorithm
transform	I-Algorithm
obey	O
the	O
same	O
equation	O
.	O
</s>
<s>
Assuming	O
uniqueness	O
of	O
the	O
solutions	O
,	O
every	O
solution	O
must	O
therefore	O
be	O
an	O
eigenfunction	B-Algorithm
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
The	O
form	O
of	O
the	O
equation	O
remains	O
unchanged	O
under	O
Fourier	B-Algorithm
transform	I-Algorithm
if	O
can	O
be	O
expanded	O
in	O
a	O
power	O
series	O
in	O
which	O
for	O
all	O
terms	O
the	O
same	O
factor	O
of	O
either	O
one	O
of	O
arises	O
from	O
the	O
factors	O
introduced	O
by	O
the	O
differentiation	B-Algorithm
rules	O
upon	O
Fourier	O
transforming	O
the	O
homogeneous	O
differential	O
equation	O
because	O
this	O
factor	O
may	O
then	O
be	O
cancelled	O
.	O
</s>
<s>
with	O
constant	O
and	O
being	O
a	O
non-constant	O
even	O
function	O
remains	O
invariant	O
in	O
form	O
when	O
applying	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
to	O
both	O
sides	O
of	O
the	O
equation	O
.	O
</s>
<s>
The	O
corresponding	O
solutions	O
provide	O
an	O
important	O
choice	O
of	O
an	O
orthonormal	B-Algorithm
basis	O
for	O
and	O
are	O
given	O
by	O
the	O
"	O
physicist	O
's	O
"	O
Hermite	O
functions	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
the	O
Hermite	O
functions	O
form	O
a	O
complete	O
orthonormal	B-Algorithm
system	O
of	O
eigenfunctions	B-Algorithm
for	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
on	O
.	O
</s>
<s>
However	O
,	O
this	O
choice	O
of	O
eigenfunctions	B-Algorithm
is	O
not	O
unique	O
.	O
</s>
<s>
Because	O
of	O
there	O
are	O
only	O
four	O
different	O
eigenvalues	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
the	O
fourth	O
roots	O
of	O
unity	O
±1	O
and	O
±	O
)	O
and	O
any	O
linear	O
combination	O
of	O
eigenfunctions	B-Algorithm
with	O
the	O
same	O
eigenvalue	O
gives	O
another	O
eigenfunction	B-Algorithm
.	O
</s>
<s>
As	O
a	O
consequence	O
of	O
this	O
,	O
it	O
is	O
possible	O
to	O
decompose	O
as	O
a	O
direct	O
sum	O
of	O
four	O
spaces	O
,	O
,	O
,	O
and	O
where	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
acts	O
on	O
simply	O
by	O
multiplication	O
by	O
.	O
</s>
<s>
the	O
Fourier	B-Algorithm
transform	I-Algorithm
can	O
be	O
represented	O
by	O
such	O
a	O
sum	O
of	O
terms	O
weighted	O
by	O
the	O
above	O
eigenvalues	O
,	O
and	O
these	O
sums	O
can	O
be	O
explicitly	O
summed	O
:	O
</s>
<s>
This	O
approach	O
to	O
define	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
was	O
first	O
proposed	O
by	O
Norbert	O
Wiener	O
.	O
</s>
<s>
Among	O
other	O
properties	O
,	O
Hermite	O
functions	O
decrease	O
exponentially	O
fast	O
in	O
both	O
frequency	O
and	O
time	O
domains	O
,	O
and	O
they	O
are	O
thus	O
used	O
to	O
define	O
a	O
generalization	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
namely	O
the	O
fractional	O
Fourier	B-Algorithm
transform	I-Algorithm
used	O
in	O
time	B-Algorithm
–	I-Algorithm
frequency	I-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
In	O
physics	O
,	O
this	O
transform	B-Algorithm
was	O
introduced	O
by	O
Edward	O
Condon	O
.	O
</s>
<s>
This	O
change	O
of	O
basis	O
functions	O
becomes	O
possible	O
because	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
a	O
unitary	B-Algorithm
transform	I-Algorithm
when	O
using	O
the	O
right	O
conventions	O
.	O
</s>
<s>
It	O
can	O
be	O
interpreted	O
as	O
the	O
generator	O
of	O
fractional	O
Fourier	B-Algorithm
transforms	I-Algorithm
for	O
arbitrary	O
values	O
of	O
,	O
and	O
of	O
the	O
conventional	O
continuous	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
for	O
the	O
particular	O
value	O
with	O
the	O
Mehler	O
kernel	B-Algorithm
implementing	O
the	O
corresponding	O
active	O
transform	B-Algorithm
.	O
</s>
<s>
Upon	O
extending	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
to	O
distributions	O
the	O
Dirac	O
comb	O
is	O
also	O
an	O
eigenfunction	B-Algorithm
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
The	O
Heisenberg	O
group	O
is	O
a	O
certain	O
group	O
of	O
unitary	B-Algorithm
operators	I-Algorithm
on	O
the	O
Hilbert	O
space	O
of	O
square	B-Algorithm
integrable	I-Algorithm
complex	O
valued	O
functions	O
on	O
the	O
real	O
line	O
,	O
generated	O
by	O
the	O
translations	O
and	O
multiplication	O
by	O
,	O
.	O
</s>
<s>
The	O
above	O
procedure	O
describes	O
not	O
only	O
the	O
group	O
structure	O
,	O
but	O
also	O
a	O
standard	O
unitary	B-Algorithm
representation	O
of	O
on	O
a	O
Hilbert	O
space	O
,	O
which	O
we	O
denote	O
by	O
.	O
</s>
<s>
This	O
operator	O
is	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
Many	O
of	O
the	O
standard	O
properties	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
are	O
immediate	O
consequences	O
of	O
this	O
more	O
general	O
framework	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
square	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
,	O
is	O
an	O
intertwiner	O
associated	O
with	O
,	O
and	O
so	O
we	O
have	O
is	O
the	O
reflection	O
of	O
the	O
original	O
function	O
.	O
</s>
<s>
Depending	O
on	O
the	O
properties	O
of	O
,	O
this	O
might	O
not	O
converge	O
off	O
the	O
real	O
axis	O
at	O
all	O
,	O
or	O
it	O
might	O
converge	O
to	O
a	O
complex	O
analytic	B-Language
function	I-Language
for	O
all	O
values	O
of	O
,	O
or	O
something	O
in	O
between	O
.	O
</s>
<s>
The	O
converse	O
is	O
false	O
and	O
it	O
is	O
not	O
known	O
how	O
to	O
characterise	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
a	O
causal	O
function	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
related	O
to	O
the	O
Laplace	O
transform	B-Algorithm
,	O
which	O
is	O
also	O
used	O
for	O
the	O
solution	O
of	O
differential	O
equations	O
and	O
the	O
analysis	O
of	O
filters	O
.	O
</s>
<s>
It	O
may	O
happen	O
that	O
a	O
function	O
for	O
which	O
the	O
Fourier	B-Algorithm
integral	I-Algorithm
does	O
not	O
converge	O
on	O
the	O
real	O
axis	O
at	O
all	O
,	O
nevertheless	O
has	O
a	O
complex	O
Fourier	B-Algorithm
transform	I-Algorithm
defined	O
in	O
some	O
region	O
of	O
the	O
complex	O
plane	O
.	O
</s>
<s>
convergent	O
for	O
all	O
,	O
is	O
the	O
two-sided	B-Algorithm
Laplace	I-Algorithm
transform	I-Algorithm
of	O
.	O
</s>
<s>
If	O
is	O
also	O
causal	O
,	O
and	O
analytical	O
,	O
then	O
:	O
Thus	O
,	O
extending	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
to	O
the	O
complex	O
domain	B-Algorithm
means	O
it	O
includes	O
the	O
Laplace	O
transform	B-Algorithm
as	O
a	O
special	O
case	O
in	O
the	O
case	O
of	O
causal	O
functions	O
—	O
but	O
with	O
the	O
change	O
of	O
variable	O
.	O
</s>
<s>
From	O
another	O
,	O
perhaps	O
more	O
classical	O
viewpoint	O
,	O
the	O
Laplace	O
transform	B-Algorithm
by	O
its	O
form	O
involves	O
an	O
additional	O
exponential	O
regulating	O
term	O
which	O
lets	O
it	O
converge	O
outside	O
of	O
the	O
imaginary	O
line	O
where	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
defined	O
.	O
</s>
<s>
However	O
,	O
they	O
do	O
admit	O
a	O
Laplace	O
domain	B-Algorithm
description	O
,	O
with	O
identical	O
half-planes	O
of	O
convergence	O
in	O
the	O
complex	O
plane	O
(	O
or	O
in	O
the	O
discrete	O
case	O
,	O
the	O
Z-plane	O
)	O
,	O
wherein	O
their	O
effects	O
cancel	O
.	O
</s>
<s>
In	O
modern	O
mathematics	O
the	O
Laplace	O
transform	B-Algorithm
is	O
conventionally	O
subsumed	O
under	O
the	O
aegis	O
Fourier	O
methods	O
.	O
</s>
<s>
This	O
theorem	O
implies	O
the	O
Mellin	B-Algorithm
inversion	I-Algorithm
formula	I-Algorithm
for	O
the	O
Laplace	O
transformation	O
,	O
</s>
<s>
for	O
any	O
,	O
where	O
is	O
the	O
Laplace	O
transform	B-Algorithm
of	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
can	O
be	O
defined	O
in	O
any	O
arbitrary	O
number	O
of	O
dimensions	O
.	O
</s>
<s>
All	O
of	O
the	O
basic	O
properties	O
listed	O
above	O
hold	O
for	O
the	O
-dimensional	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
as	O
do	O
Plancherel	O
's	O
and	O
Parseval	O
's	O
theorem	O
.	O
</s>
<s>
When	O
the	O
function	O
is	O
integrable	O
,	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
still	O
uniformly	O
continuous	O
and	O
the	O
Riemann	O
–	O
Lebesgue	O
lemma	O
holds	O
.	O
</s>
<s>
Generally	O
speaking	O
,	O
the	O
more	O
concentrated	O
is	O
,	O
the	O
more	O
spread	O
out	O
its	O
Fourier	B-Algorithm
transform	I-Algorithm
must	O
be	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
scaling	O
property	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
may	O
be	O
seen	O
as	O
saying	O
:	O
if	O
we	O
squeeze	O
a	O
function	O
in	O
,	O
its	O
Fourier	B-Algorithm
transform	I-Algorithm
stretches	O
out	O
in	O
.	O
</s>
<s>
It	O
is	O
not	O
possible	O
to	O
arbitrarily	O
concentrate	O
both	O
a	O
function	O
and	O
its	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
The	O
trade-off	O
between	O
the	O
compaction	O
of	O
a	O
function	O
and	O
its	O
Fourier	B-Algorithm
transform	I-Algorithm
can	O
be	O
formalized	O
in	O
the	O
form	O
of	O
an	O
uncertainty	O
principle	O
by	O
viewing	O
a	O
function	O
and	O
its	O
Fourier	B-Algorithm
transform	I-Algorithm
as	O
conjugate	O
variables	O
with	O
respect	O
to	O
the	O
symplectic	O
form	O
on	O
the	O
time	O
–	O
frequency	O
domain	B-Algorithm
:	O
from	O
the	O
point	O
of	O
view	O
of	O
the	O
linear	B-Algorithm
canonical	I-Algorithm
transformation	I-Algorithm
,	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
rotation	O
by	O
90°	O
in	O
the	O
time	O
–	O
frequency	O
domain	B-Algorithm
,	O
and	O
preserves	O
the	O
symplectic	O
form	O
.	O
</s>
<s>
Suppose	O
is	O
an	O
integrable	O
and	O
square-integrable	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
In	O
other	O
words	O
,	O
where	O
is	O
a	O
(	O
normalized	O
)	O
Gaussian	O
function	O
with	O
variance	O
,	O
centered	O
at	O
zero	O
,	O
and	O
its	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
a	O
Gaussian	O
function	O
with	O
variance	O
.	O
</s>
<s>
In	O
quantum	O
mechanics	O
,	O
the	O
momentum	B-Algorithm
and	O
position	O
wave	O
functions	O
are	O
Fourier	B-Algorithm
transform	I-Algorithm
pairs	O
,	O
to	O
within	O
a	O
factor	O
of	O
Planck	O
's	O
constant	O
.	O
</s>
<s>
Fourier	O
's	O
original	O
formulation	O
of	O
the	O
transform	B-Algorithm
did	O
not	O
use	O
complex	O
numbers	O
,	O
but	O
rather	O
sines	O
and	O
cosines	O
.	O
</s>
<s>
This	O
is	O
called	O
an	O
expansion	O
as	O
a	O
trigonometric	O
integral	O
,	O
or	O
a	O
Fourier	B-Algorithm
integral	I-Algorithm
expansion	O
.	O
</s>
<s>
The	O
coefficient	O
functions	O
and	O
can	O
be	O
found	O
by	O
using	O
variants	O
of	O
the	O
Fourier	O
cosine	O
transform	B-Algorithm
and	O
the	O
Fourier	O
sine	O
transform	B-Algorithm
(	O
the	O
normalisations	O
are	O
,	O
again	O
,	O
not	O
standardised	O
)	O
:	O
</s>
<s>
Older	O
literature	O
refers	O
to	O
the	O
two	O
transform	B-Algorithm
functions	O
,	O
the	O
Fourier	O
cosine	O
transform	B-Algorithm
,	O
,	O
and	O
the	O
Fourier	O
sine	O
transform	B-Algorithm
,	O
.	O
</s>
<s>
The	O
space	O
is	O
then	O
a	O
direct	O
sum	O
of	O
the	O
spaces	O
and	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
maps	O
each	O
space	O
to	O
itself	O
and	O
is	O
possible	O
to	O
characterize	O
the	O
action	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
on	O
each	O
space	O
.	O
</s>
<s>
When	O
this	O
gives	O
a	O
useful	O
formula	O
for	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
a	O
radial	O
function	O
.	O
</s>
<s>
This	O
is	O
essentially	O
the	O
Hankel	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
Moreover	O
,	O
there	O
is	O
a	O
simple	O
recursion	O
relating	O
the	O
cases	O
and	O
allowing	O
to	O
compute	O
,	O
e.g.	O
,	O
the	O
three-dimensional	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
a	O
radial	O
function	O
from	O
the	O
one-dimensional	O
one	O
.	O
</s>
<s>
In	O
higher	O
dimensions	O
it	O
becomes	O
interesting	O
to	O
study	O
restriction	O
problems	O
for	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
an	O
integrable	O
function	O
is	O
continuous	O
and	O
the	O
restriction	O
of	O
this	O
function	O
to	O
any	O
set	O
is	O
defined	O
.	O
</s>
<s>
But	O
for	O
a	O
square-integrable	B-Algorithm
function	I-Algorithm
the	O
Fourier	B-Algorithm
transform	I-Algorithm
could	O
be	O
a	O
general	O
class	O
of	O
square	B-Algorithm
integrable	I-Algorithm
functions	I-Algorithm
.	O
</s>
<s>
As	O
such	O
,	O
the	O
restriction	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
an	O
function	O
cannot	O
be	O
defined	O
on	O
sets	O
of	O
measure	O
0	O
.	O
</s>
<s>
Surprisingly	O
,	O
it	O
is	O
possible	O
in	O
some	O
cases	O
to	O
define	O
the	O
restriction	O
of	O
a	O
Fourier	B-Algorithm
transform	I-Algorithm
to	O
a	O
set	O
,	O
provided	O
has	O
non-zero	O
curvature	O
.	O
</s>
<s>
In	O
this	O
case	O
the	O
Tomas	O
–	O
Stein	O
restriction	O
theorem	O
states	O
that	O
the	O
restriction	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
to	O
the	O
unit	O
sphere	O
in	O
is	O
a	O
bounded	O
operator	O
on	O
provided	O
.	O
</s>
<s>
One	O
notable	O
difference	O
between	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
in	O
1	O
dimension	O
versus	O
higher	O
dimensions	O
concerns	O
the	O
partial	O
sum	O
operator	O
.	O
</s>
<s>
For	O
and	O
,	O
if	O
one	O
takes	O
,	O
then	O
converges	O
to	O
in	O
as	O
tends	O
to	O
infinity	O
,	O
by	O
the	O
boundedness	O
of	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
a	O
bounded	O
operator	O
.	O
</s>
<s>
Indeed	O
,	O
it	O
equals	O
1	O
,	O
which	O
can	O
be	O
seen	O
,	O
for	O
example	O
,	O
from	O
the	O
transform	B-Algorithm
of	O
the	O
rect	O
function	O
.	O
</s>
<s>
Since	O
compactly	O
supported	O
smooth	O
functions	O
are	O
integrable	O
and	O
dense	O
in	O
,	O
the	O
Plancherel	O
theorem	O
allows	O
us	O
to	O
extend	O
the	O
definition	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
to	O
general	O
functions	O
in	O
by	O
continuity	O
arguments	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
in	O
is	O
no	O
longer	O
given	O
by	O
an	O
ordinary	O
Lebesgue	O
integral	O
,	O
although	O
it	O
can	O
be	O
computed	O
by	O
an	O
improper	O
integral	O
,	O
here	O
meaning	O
that	O
for	O
an	O
function	O
,	O
</s>
<s>
(	O
More	O
generally	O
,	O
you	O
can	O
take	O
a	O
sequence	O
of	O
functions	O
that	O
are	O
in	O
the	O
intersection	O
of	O
and	O
and	O
that	O
converges	O
to	O
in	O
the	O
-norm	O
,	O
and	O
define	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
as	O
the	O
-limit	O
of	O
the	O
Fourier	B-Algorithm
transforms	I-Algorithm
of	O
these	O
functions	O
.	O
)	O
</s>
<s>
Many	O
of	O
the	O
properties	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
in	O
carry	O
over	O
to	O
,	O
by	O
a	O
suitable	O
limiting	O
argument	O
.	O
</s>
<s>
Furthermore	O
,	O
is	O
a	O
unitary	B-Algorithm
operator	I-Algorithm
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
image	O
of	O
is	O
itself	O
under	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
The	O
definition	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
can	O
be	O
extended	O
to	O
functions	O
in	O
for	O
by	O
decomposing	O
such	O
functions	O
into	O
a	O
fat	O
tail	O
part	O
in	O
plus	O
a	O
fat	O
body	O
part	O
in	O
.	O
</s>
<s>
In	O
each	O
of	O
these	O
spaces	O
,	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
a	O
function	O
in	O
is	O
in	O
,	O
where	O
is	O
the	O
Hölder	B-Algorithm
conjugate	I-Algorithm
of	O
(	O
by	O
the	O
Hausdorff	O
–	O
Young	O
inequality	O
)	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
functions	O
in	O
for	O
the	O
range	B-Algorithm
requires	O
the	O
study	O
of	O
distributions	O
.	O
</s>
<s>
In	O
fact	O
,	O
it	O
can	O
be	O
shown	O
that	O
there	O
are	O
functions	O
in	O
with	O
so	O
that	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
not	O
defined	O
as	O
a	O
function	O
.	O
</s>
<s>
One	O
might	O
consider	O
enlarging	O
the	O
domain	B-Algorithm
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
from	O
by	O
considering	O
generalized	O
functions	O
,	O
or	O
distributions	O
.	O
</s>
<s>
A	O
distribution	O
on	O
is	O
a	O
continuous	O
linear	B-Algorithm
functional	I-Algorithm
on	O
the	O
space	O
of	O
compactly	O
supported	O
smooth	O
functions	O
,	O
equipped	O
with	O
a	O
suitable	O
topology	O
.	O
</s>
<s>
The	O
strategy	O
is	O
then	O
to	O
consider	O
the	O
action	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
on	O
and	O
pass	O
to	O
distributions	O
by	O
duality	O
.	O
</s>
<s>
The	O
obstruction	O
to	O
doing	O
this	O
is	O
that	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
does	O
not	O
map	O
to	O
.	O
</s>
<s>
In	O
fact	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
an	O
element	O
in	O
can	O
not	O
vanish	O
on	O
an	O
open	O
set	O
;	O
see	O
the	O
above	O
discussion	O
on	O
the	O
uncertainty	O
principle	O
.	O
</s>
<s>
The	O
right	O
space	O
here	O
is	O
the	O
slightly	O
larger	O
space	O
of	O
Schwartz	B-Algorithm
functions	I-Algorithm
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
an	O
automorphism	O
on	O
the	O
Schwartz	B-Algorithm
space	I-Algorithm
,	O
as	O
a	O
topological	O
vector	O
space	O
,	O
and	O
thus	O
induces	O
an	O
automorphism	O
on	O
its	O
dual	O
,	O
the	O
space	O
of	O
tempered	O
distributions	O
.	O
</s>
<s>
For	O
the	O
definition	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
a	O
tempered	O
distribution	O
,	O
let	O
and	O
be	O
integrable	O
functions	O
,	O
and	O
let	O
and	O
be	O
their	O
Fourier	B-Algorithm
transforms	I-Algorithm
respectively	O
.	O
</s>
<s>
Then	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
obeys	O
the	O
following	O
multiplication	O
formula	O
,	O
</s>
<s>
for	O
all	O
Schwartz	B-Algorithm
functions	I-Algorithm
.	O
</s>
<s>
for	O
all	O
Schwartz	B-Algorithm
functions	I-Algorithm
.	O
</s>
<s>
Extending	O
this	O
to	O
all	O
tempered	O
distributions	O
gives	O
the	O
general	O
definition	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
Distributions	O
can	O
be	O
differentiated	O
and	O
the	O
above-mentioned	O
compatibility	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
with	O
differentiation	B-Algorithm
and	O
convolution	B-Language
remains	O
true	O
for	O
tempered	O
distributions	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
a	O
finite	O
Borel	O
measure	O
on	O
is	O
given	O
by	O
:	O
</s>
<s>
This	O
transform	B-Algorithm
continues	O
to	O
enjoy	O
many	O
of	O
the	O
properties	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
integrable	O
functions	O
.	O
</s>
<s>
In	O
the	O
case	O
that	O
,	O
then	O
the	O
formula	O
above	O
reduces	O
to	O
the	O
usual	O
definition	O
for	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
.	O
</s>
<s>
In	O
the	O
case	O
that	O
is	O
the	O
probability	O
distribution	O
associated	O
to	O
a	O
random	O
variable	O
,	O
the	O
Fourier	O
–	O
Stieltjes	O
transform	B-Algorithm
is	O
closely	O
related	O
to	O
the	O
characteristic	O
function	O
,	O
but	O
the	O
typical	O
conventions	O
in	O
probability	O
theory	O
take	O
instead	O
of	O
.	O
</s>
<s>
In	O
the	O
case	O
when	O
the	O
distribution	O
has	O
a	O
probability	O
density	O
function	O
this	O
definition	O
reduces	O
to	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
applied	O
to	O
the	O
probability	O
density	O
function	O
,	O
again	O
with	O
a	O
different	O
choice	O
of	O
constants	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
may	O
be	O
used	O
to	O
give	O
a	O
characterization	O
of	O
measures	O
.	O
</s>
<s>
Bochner	O
's	O
theorem	O
characterizes	O
which	O
functions	O
may	O
arise	O
as	O
the	O
Fourier	O
–	O
Stieltjes	O
transform	B-Algorithm
of	O
a	O
positive	O
measure	O
on	O
the	O
circle	O
.	O
</s>
<s>
Its	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
a	O
constant	O
function	O
(	O
whose	O
specific	O
value	O
depends	O
upon	O
the	O
form	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
used	O
)	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
may	O
be	O
generalized	O
to	O
any	O
locally	O
compact	O
abelian	O
group	O
.	O
</s>
<s>
If	O
is	O
a	O
locally	O
compact	O
abelian	O
group	O
,	O
it	O
has	O
a	O
translation	B-Algorithm
invariant	O
measure	O
,	O
called	O
Haar	O
measure	O
.	O
</s>
<s>
one-dimensional	O
,	O
unitary	B-Algorithm
representations	O
are	O
called	O
its	O
characters	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
on	O
is	O
an	O
example	O
;	O
here	O
is	O
a	O
locally	O
compact	O
abelian	O
group	O
,	O
and	O
the	O
Haar	O
measure	O
on	O
can	O
be	O
thought	O
of	O
as	O
the	O
Lebesgue	O
measure	O
on	O
[	O
0	O
,	O
1	O
)	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
representation	O
of	O
finite	O
group	O
,	O
the	O
character	O
table	O
of	O
the	O
group	O
are	O
rows	O
of	O
vectors	O
such	O
that	O
each	O
row	O
is	O
the	O
character	O
of	O
one	O
irreducible	O
representation	O
of	O
,	O
and	O
these	O
vectors	O
form	O
an	O
orthonormal	B-Algorithm
basis	O
of	O
the	O
space	O
of	O
class	O
functions	O
that	O
map	O
from	O
to	O
by	O
Schur	O
's	O
lemma	O
.	O
</s>
<s>
Now	O
the	O
group	O
is	O
no	O
longer	O
finite	O
but	O
still	O
compact	O
,	O
and	O
it	O
preserves	O
the	O
orthonormality	B-Algorithm
of	O
character	O
table	O
.	O
</s>
<s>
The	O
sequence	O
is	O
an	O
orthonormal	B-Algorithm
basis	O
of	O
the	O
space	O
of	O
class	O
functions	O
.	O
</s>
<s>
The	O
Pontriagin	O
dual	O
is	O
and	O
for	O
,	O
is	O
its	O
Fourier	B-Algorithm
transform	I-Algorithm
for	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
also	O
a	O
special	O
case	O
of	O
Gelfand	O
transform	B-Algorithm
.	O
</s>
<s>
With	O
convolution	B-Language
as	O
multiplication	O
,	O
is	O
an	O
abelian	O
Banach	O
algebra	O
.	O
</s>
<s>
Given	O
any	O
abelian	O
-algebra	O
,	O
the	O
Gelfand	O
transform	B-Algorithm
gives	O
an	O
isomorphism	O
between	O
and	O
,	O
where	O
is	O
the	O
multiplicative	O
linear	B-Algorithm
functionals	I-Algorithm
,	O
i.e.	O
</s>
<s>
It	O
turns	O
out	O
that	O
the	O
multiplicative	O
linear	B-Algorithm
functionals	I-Algorithm
of	O
,	O
after	O
suitable	O
identification	O
,	O
are	O
exactly	O
the	O
characters	O
of	O
,	O
and	O
the	O
Gelfand	O
transform	B-Algorithm
,	O
when	O
restricted	O
to	O
the	O
dense	O
subset	O
is	O
the	O
Fourier	O
–	O
Pontryagin	O
transform	B-Algorithm
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
can	O
also	O
be	O
defined	O
for	O
functions	O
on	O
a	O
non-abelian	O
group	O
,	O
provided	O
that	O
the	O
group	O
is	O
compact	O
.	O
</s>
<s>
Removing	O
the	O
assumption	O
that	O
the	O
underlying	O
group	O
is	O
abelian	O
,	O
irreducible	O
unitary	B-Algorithm
representations	O
need	O
not	O
always	O
be	O
one-dimensional	O
.	O
</s>
<s>
This	O
means	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
on	O
a	O
non-abelian	O
group	O
takes	O
values	O
as	O
Hilbert	O
space	O
operators	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
on	O
compact	O
groups	O
is	O
a	O
major	O
tool	O
in	O
representation	O
theory	O
and	O
non-commutative	O
harmonic	O
analysis	O
.	O
</s>
<s>
Let	O
denote	O
the	O
collection	O
of	O
all	O
isomorphism	O
classes	O
of	O
finite-dimensional	O
irreducible	O
unitary	B-Algorithm
representations	O
,	O
along	O
with	O
a	O
definite	O
choice	O
of	O
representation	O
on	O
the	O
Hilbert	O
space	O
of	O
finite	O
dimension	O
for	O
each	O
.	O
</s>
<s>
for	O
some	O
,	O
one	O
identifies	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
with	O
the	O
Fourier	O
–	O
Stieltjes	O
transform	B-Algorithm
of	O
.	O
</s>
<s>
The	O
"	O
convolution	B-Language
theorem	O
"	O
asserts	O
that	O
,	O
furthermore	O
,	O
this	O
isomorphism	O
of	O
Banach	O
spaces	O
is	O
in	O
fact	O
an	O
isometric	O
isomorphism	O
of	O
C*	B-Algorithm
-algebras	I-Algorithm
into	O
a	O
subspace	O
of	O
.	O
</s>
<s>
The	O
generalization	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
to	O
the	O
noncommutative	O
situation	O
has	O
also	O
in	O
part	O
contributed	O
to	O
the	O
development	O
of	O
noncommutative	O
geometry	O
.	O
</s>
<s>
In	O
this	O
context	O
,	O
a	O
categorical	O
generalization	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
to	O
noncommutative	O
groups	O
is	O
Tannaka	O
–	O
Krein	O
duality	O
,	O
which	O
replaces	O
the	O
group	O
of	O
characters	O
with	O
the	O
category	O
of	O
representations	O
.	O
</s>
<s>
In	O
signal	O
processing	O
terms	O
,	O
a	O
function	O
(	O
of	O
time	O
)	O
is	O
a	O
representation	O
of	O
a	O
signal	O
with	O
perfect	O
time	O
resolution	O
,	O
but	O
no	O
frequency	O
information	O
,	O
while	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
has	O
perfect	O
frequency	O
resolution	O
,	O
but	O
no	O
time	O
information	O
:	O
the	O
magnitude	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
at	O
a	O
point	O
is	O
how	O
much	O
frequency	O
content	O
there	O
is	O
,	O
but	O
location	O
is	O
only	O
given	O
by	O
phase	O
(	O
argument	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
at	O
a	O
point	O
)	O
,	O
and	O
standing	O
waves	O
are	O
not	O
localized	O
in	O
time	O
–	O
a	O
sine	O
wave	O
continues	O
out	O
to	O
infinity	O
,	O
without	O
decaying	O
.	O
</s>
<s>
This	O
limits	O
the	O
usefulness	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
for	O
analyzing	O
signals	O
that	O
are	O
localized	O
in	O
time	O
,	O
notably	O
transients	O
,	O
or	O
any	O
signal	O
of	O
finite	O
extent	O
.	O
</s>
<s>
As	O
alternatives	O
to	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
in	O
time	B-Algorithm
–	I-Algorithm
frequency	I-Algorithm
analysis	I-Algorithm
,	O
one	O
uses	O
time	O
–	O
frequency	B-Algorithm
transforms	I-Algorithm
or	O
time	O
–	O
frequency	O
distributions	O
to	O
represent	O
signals	O
in	O
a	O
form	O
that	O
has	O
some	O
time	O
information	O
and	O
some	O
frequency	O
information	O
–	O
by	O
the	O
uncertainty	O
principle	O
,	O
there	O
is	O
a	O
trade-off	O
between	O
these	O
.	O
</s>
<s>
These	O
can	O
be	O
generalizations	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
such	O
as	O
the	O
short-time	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
or	O
fractional	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
or	O
other	O
functions	O
to	O
represent	O
signals	O
,	O
as	O
in	O
wavelet	B-Algorithm
transforms	I-Algorithm
and	O
chirplet	B-Algorithm
transforms	I-Algorithm
,	O
with	O
the	O
wavelet	O
analog	O
of	O
the	O
(	O
continuous	O
)	O
Fourier	B-Algorithm
transform	I-Algorithm
being	O
the	O
continuous	B-Algorithm
wavelet	I-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
Linear	O
operations	O
performed	O
in	O
one	O
domain	B-Algorithm
(	O
time	O
or	O
frequency	O
)	O
have	O
corresponding	O
operations	O
in	O
the	O
other	O
domain	B-Algorithm
,	O
which	O
are	O
sometimes	O
easier	O
to	O
perform	O
.	O
</s>
<s>
The	O
operation	O
of	O
differentiation	B-Algorithm
in	O
the	O
time	O
domain	B-Algorithm
corresponds	O
to	O
multiplication	O
by	O
the	O
frequency	O
,	O
so	O
some	O
differential	O
equations	O
are	O
easier	O
to	O
analyze	O
in	O
the	O
frequency	O
domain	B-Algorithm
.	O
</s>
<s>
Also	O
,	O
convolution	B-Language
in	O
the	O
time	O
domain	B-Algorithm
corresponds	O
to	O
ordinary	O
multiplication	O
in	O
the	O
frequency	O
domain	B-Algorithm
(	O
see	O
Convolution	B-Language
theorem	O
)	O
.	O
</s>
<s>
After	O
performing	O
the	O
desired	O
operations	O
,	O
transformation	O
of	O
the	O
result	O
can	O
be	O
made	O
back	O
to	O
the	O
time	O
domain	B-Algorithm
.	O
</s>
<s>
Perhaps	O
the	O
most	O
important	O
use	O
of	O
the	O
Fourier	B-Algorithm
transformation	I-Algorithm
is	O
to	O
solve	O
partial	O
differential	O
equations	O
.	O
</s>
<s>
It	O
is	O
easier	O
to	O
find	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
the	O
solution	O
than	O
to	O
find	O
the	O
solution	O
directly	O
.	O
</s>
<s>
This	O
is	O
because	O
the	O
Fourier	B-Algorithm
transformation	I-Algorithm
takes	O
differentiation	B-Algorithm
into	O
multiplication	O
by	O
the	O
Fourier-dual	O
variable	O
,	O
and	O
so	O
a	O
partial	O
differential	O
equation	O
applied	O
to	O
the	O
original	O
function	O
is	O
transformed	O
into	O
multiplication	O
by	O
polynomial	O
functions	O
of	O
the	O
dual	O
variables	O
applied	O
to	O
the	O
transformed	O
function	O
.	O
</s>
<s>
After	O
is	O
determined	O
,	O
we	O
can	O
apply	O
the	O
inverse	O
Fourier	B-Algorithm
transformation	I-Algorithm
to	O
find	O
.	O
</s>
<s>
In	O
fact	O
,	O
this	O
is	O
the	O
real	O
inverse	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
and	O
in	O
the	O
variable	O
.	O
</s>
<s>
But	O
this	O
integral	O
was	O
in	O
the	O
form	O
of	O
a	O
Fourier	B-Algorithm
integral	I-Algorithm
.	O
</s>
<s>
But	O
these	O
expressions	O
also	O
took	O
the	O
form	O
of	O
a	O
Fourier	B-Algorithm
integral	I-Algorithm
because	O
of	O
the	O
properties	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
a	O
derivative	B-Algorithm
.	O
</s>
<s>
The	O
last	O
step	O
was	O
to	O
exploit	O
Fourier	O
inversion	O
by	O
applying	O
the	O
Fourier	B-Algorithm
transformation	I-Algorithm
to	O
both	O
sides	O
,	O
thus	O
obtaining	O
expressions	O
for	O
the	O
coefficient	O
functions	O
and	O
in	O
terms	O
of	O
the	O
given	O
boundary	O
conditions	O
and	O
.	O
</s>
<s>
Since	O
there	O
are	O
two	O
variables	O
,	O
we	O
will	O
use	O
the	O
Fourier	B-Algorithm
transformation	I-Algorithm
in	O
both	O
and	O
rather	O
than	O
operate	O
as	O
Fourier	O
did	O
,	O
who	O
only	O
transformed	O
in	O
the	O
spatial	O
variables	O
.	O
</s>
<s>
But	O
it	O
will	O
be	O
bounded	O
and	O
so	O
its	O
Fourier	B-Algorithm
transform	I-Algorithm
can	O
be	O
defined	O
as	O
a	O
distribution	O
.	O
</s>
<s>
The	O
operational	O
properties	O
of	O
the	O
Fourier	B-Algorithm
transformation	I-Algorithm
that	O
are	O
relevant	O
to	O
this	O
equation	O
are	O
that	O
it	O
takes	O
differentiation	B-Algorithm
in	O
to	O
multiplication	O
by	O
and	O
differentiation	B-Algorithm
with	O
respect	O
to	O
to	O
multiplication	O
by	O
where	O
is	O
the	O
frequency	O
.	O
</s>
<s>
Now	O
,	O
as	O
before	O
,	O
applying	O
the	O
one-variable	O
Fourier	B-Algorithm
transformation	I-Algorithm
in	O
the	O
variable	O
to	O
these	O
functions	O
of	O
yields	O
two	O
equations	O
in	O
the	O
two	O
unknown	O
distributions	O
(	O
which	O
can	O
be	O
taken	O
to	O
be	O
ordinary	O
functions	O
if	O
the	O
boundary	O
conditions	O
are	O
or	O
)	O
.	O
</s>
<s>
From	O
a	O
calculational	O
point	O
of	O
view	O
,	O
the	O
drawback	O
of	O
course	O
is	O
that	O
one	O
must	O
first	O
calculate	O
the	O
Fourier	B-Algorithm
transforms	I-Algorithm
of	O
the	O
boundary	O
conditions	O
,	O
then	O
assemble	O
the	O
solution	O
from	O
these	O
,	O
and	O
then	O
calculate	O
an	O
inverse	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
The	O
twentieth	O
century	O
has	O
seen	O
the	O
extension	O
of	O
these	O
methods	O
to	O
all	O
linear	O
partial	O
differential	O
equations	O
with	O
polynomial	O
coefficients	O
,	O
and	O
by	O
extending	O
the	O
notion	O
of	O
Fourier	B-Algorithm
transformation	I-Algorithm
to	O
include	O
Fourier	B-Algorithm
integral	I-Algorithm
operators	O
,	O
some	O
non-linear	O
equations	O
as	O
well	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
also	O
used	O
in	O
nuclear	O
magnetic	O
resonance	O
(	O
NMR	O
)	O
and	O
in	O
other	O
kinds	O
of	O
spectroscopy	O
,	O
e.g.	O
</s>
<s>
In	O
NMR	O
an	O
exponentially	O
shaped	O
free	O
induction	O
decay	O
(	O
FID	O
)	O
signal	O
is	O
acquired	O
in	O
the	O
time	O
domain	B-Algorithm
and	O
Fourier-transformed	O
to	O
a	O
Lorentzian	O
line-shape	O
in	O
the	O
frequency	O
domain	B-Algorithm
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
also	O
used	O
in	O
magnetic	O
resonance	O
imaging	O
(	O
MRI	O
)	O
and	O
mass	O
spectrometry	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
useful	O
in	O
quantum	O
mechanics	O
in	O
at	O
least	O
two	O
different	O
ways	O
.	O
</s>
<s>
For	O
example	O
,	O
in	O
one	O
dimension	O
,	O
the	O
spatial	O
variable	O
of	O
,	O
say	O
,	O
a	O
particle	O
,	O
can	O
only	O
be	O
measured	O
by	O
the	O
quantum	O
mechanical	O
"	O
position	O
operator	O
"	O
at	O
the	O
cost	O
of	O
losing	O
information	O
about	O
the	O
momentum	B-Algorithm
of	O
the	O
particle	O
.	O
</s>
<s>
Physically	O
realisable	O
states	O
are	O
,	O
and	O
so	O
by	O
the	O
Plancherel	O
theorem	O
,	O
their	O
Fourier	B-Algorithm
transforms	I-Algorithm
are	O
also	O
.	O
</s>
<s>
(	O
Note	O
that	O
since	O
is	O
in	O
units	O
of	O
distance	O
and	O
is	O
in	O
units	O
of	O
momentum	B-Algorithm
,	O
the	O
presence	O
of	O
Planck	O
's	O
constant	O
in	O
the	O
exponent	O
makes	O
the	O
exponent	O
dimensionless	O
,	O
as	O
it	O
should	O
be	O
.	O
)	O
</s>
<s>
Therefore	O
,	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
can	O
be	O
used	O
to	O
pass	O
from	O
one	O
way	O
of	O
representing	O
the	O
state	O
of	O
the	O
particle	O
,	O
by	O
a	O
wave	O
function	O
of	O
position	O
,	O
to	O
another	O
way	O
of	O
representing	O
the	O
state	O
of	O
the	O
particle	O
:	O
by	O
a	O
wave	O
function	O
of	O
momentum	B-Algorithm
.	O
</s>
<s>
Being	O
able	O
to	O
transform	B-Algorithm
states	O
from	O
one	O
representation	O
to	O
another	O
by	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
not	O
only	O
convenient	O
but	O
also	O
the	O
underlying	O
reason	O
of	O
the	O
Heisenberg	O
uncertainty	O
principle	O
.	O
</s>
<s>
The	O
other	O
use	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
in	O
both	O
quantum	O
mechanics	O
and	O
quantum	O
field	O
theory	O
is	O
to	O
solve	O
the	O
applicable	O
wave	O
equation	O
.	O
</s>
<s>
This	O
is	O
of	O
great	O
use	O
in	O
quantum	O
field	O
theory	O
:	O
each	O
separate	O
Fourier	B-Algorithm
component	I-Algorithm
of	O
a	O
wave	O
can	O
be	O
treated	O
as	O
a	O
separate	O
harmonic	O
oscillator	O
and	O
then	O
quantized	O
,	O
a	O
procedure	O
known	O
as	O
"	O
second	O
quantization	O
"	O
.	O
</s>
<s>
Finally	O
,	O
the	O
number	O
operator	O
of	O
the	O
quantum	O
harmonic	O
oscillator	O
can	O
be	O
interpreted	O
,	O
for	O
example	O
via	O
the	O
Mehler	O
kernel	B-Algorithm
,	O
as	O
the	O
generator	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
used	O
for	O
the	O
spectral	O
analysis	O
of	O
time-series	O
.	O
</s>
<s>
The	O
subject	O
of	O
statistical	O
signal	O
processing	O
does	O
not	O
,	O
however	O
,	O
usually	O
apply	O
the	O
Fourier	B-Algorithm
transformation	I-Algorithm
to	O
the	O
signal	O
itself	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
such	O
a	O
function	O
does	O
not	O
exist	O
in	O
the	O
usual	O
sense	O
,	O
and	O
it	O
has	O
been	O
found	O
more	O
useful	O
for	O
the	O
analysis	O
of	O
signals	O
to	O
instead	O
take	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
its	O
autocorrelation	O
function	O
.	O
</s>
<s>
It	O
possesses	O
a	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
</s>
<s>
This	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
called	O
the	O
power	O
spectral	B-Algorithm
density	I-Algorithm
function	I-Algorithm
of	O
.	O
</s>
<s>
This	O
process	O
is	O
called	O
the	O
spectral	O
analysis	O
of	O
time-series	O
and	O
is	O
analogous	O
to	O
the	O
usual	O
analysis	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
of	O
data	O
that	O
is	O
not	O
a	O
time-series	O
(	O
ANOVA	B-General_Concept
)	O
.	O
</s>
<s>
The	O
power	O
spectrum	O
ignores	O
all	O
phase	O
relations	O
,	O
which	O
is	O
good	O
enough	O
for	O
many	O
purposes	O
,	O
but	O
for	O
video	O
signals	O
other	O
types	O
of	O
spectral	O
analysis	O
must	O
also	O
be	O
employed	O
,	O
still	O
using	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
as	O
a	O
tool	O
.	O
</s>
<s>
Denoting	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
by	O
a	O
capital	O
letter	O
corresponding	O
to	O
the	O
letter	O
of	O
function	O
being	O
transformed	O
(	O
such	O
as	O
and	O
)	O
is	O
especially	O
common	O
in	O
the	O
sciences	O
and	O
engineering	O
.	O
</s>
<s>
In	O
electronics	O
,	O
omega	B-General_Concept
(	O
)	O
is	O
often	O
used	O
instead	O
of	O
due	O
to	O
its	O
interpretation	O
as	O
angular	O
frequency	O
,	O
sometimes	O
it	O
is	O
written	O
as	O
,	O
where	O
is	O
the	O
imaginary	O
unit	O
,	O
to	O
indicate	O
its	O
relationship	O
with	O
the	O
Laplace	O
transform	B-Algorithm
,	O
and	O
sometimes	O
it	O
is	O
written	O
informally	O
as	O
in	O
order	O
to	O
use	O
ordinary	O
frequency	O
.	O
</s>
<s>
In	O
some	O
contexts	O
such	O
as	O
particle	O
physics	O
,	O
the	O
same	O
symbol	O
may	O
be	O
used	O
for	O
both	O
for	O
a	O
function	O
as	O
well	O
as	O
it	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
with	O
the	O
two	O
only	O
distinguished	O
by	O
their	O
argument	O
:	O
would	O
refer	O
to	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
because	O
of	O
the	O
momentum	B-Algorithm
argument	O
,	O
while	O
would	O
refer	O
to	O
the	O
original	O
function	O
because	O
of	O
the	O
positional	O
argument	O
.	O
</s>
<s>
Although	O
tildes	O
may	O
be	O
used	O
as	O
in	O
to	O
indicate	O
Fourier	B-Algorithm
transforms	I-Algorithm
,	O
tildes	O
may	O
also	O
be	O
used	O
to	O
indicate	O
a	O
modification	O
of	O
a	O
quantity	O
with	O
a	O
more	O
Lorentz	O
invariant	O
form	O
,	O
such	O
as	O
,	O
so	O
care	O
must	O
be	O
taken	O
.	O
</s>
<s>
Similarly	O
,	O
often	O
denotes	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
of	O
.	O
</s>
<s>
Then	O
the	O
inverse	O
transform	B-Algorithm
can	O
be	O
written	O
:	O
</s>
<s>
Each	O
component	O
is	O
a	O
complex	O
sinusoid	O
of	O
the	O
form	O
whose	O
amplitude	B-Application
is	O
and	O
whose	O
initial	O
phase	O
angle	O
(	O
at	O
)	O
is	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
may	O
be	O
thought	O
of	O
as	O
a	O
mapping	O
on	O
function	O
spaces	O
.	O
</s>
<s>
This	O
mapping	O
is	O
here	O
denoted	O
and	O
is	O
used	O
to	O
denote	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
the	O
function	O
.	O
</s>
<s>
This	O
mapping	O
is	O
linear	O
,	O
which	O
means	O
that	O
can	O
also	O
be	O
seen	O
as	O
a	O
linear	O
transformation	O
on	O
the	O
function	O
space	O
and	O
implies	O
that	O
the	O
standard	O
notation	O
in	O
linear	B-Language
algebra	I-Language
of	O
applying	O
a	O
linear	O
transformation	O
to	O
a	O
vector	O
(	O
here	O
the	O
function	O
)	O
can	O
be	O
used	O
to	O
write	O
instead	O
of	O
.	O
</s>
<s>
Since	O
the	O
result	O
of	O
applying	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
again	O
a	O
function	O
,	O
we	O
can	O
be	O
interested	O
in	O
the	O
value	O
of	O
this	O
function	O
evaluated	O
at	O
the	O
value	O
for	O
its	O
variable	O
,	O
and	O
this	O
is	O
denoted	O
either	O
as	O
or	O
as	O
.	O
</s>
<s>
This	O
means	O
that	O
a	O
notation	O
like	O
formally	O
can	O
be	O
interpreted	O
as	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
the	O
values	O
of	O
at	O
.	O
</s>
<s>
is	O
used	O
to	O
express	O
the	O
shift	O
property	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
can	O
also	O
be	O
written	O
in	O
terms	O
of	O
angular	O
frequency	O
:	O
</s>
<s>
Under	O
this	O
convention	O
,	O
the	O
inverse	O
transform	B-Algorithm
becomes	O
:	O
</s>
<s>
Unlike	O
the	O
convention	O
followed	O
in	O
this	O
article	O
,	O
when	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
defined	O
this	O
way	O
,	O
it	O
is	O
no	O
longer	O
a	O
unitary	B-Algorithm
transformation	I-Algorithm
on	O
.	O
</s>
<s>
There	O
is	O
also	O
less	O
symmetry	O
between	O
the	O
formulas	O
for	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
and	O
its	O
inverse	O
.	O
</s>
<s>
Another	O
convention	O
is	O
to	O
split	O
the	O
factor	O
of	O
evenly	O
between	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
and	O
its	O
inverse	O
,	O
which	O
leads	O
to	O
definitions	O
:	O
</s>
<s>
Under	O
this	O
convention	O
,	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
again	O
a	O
unitary	B-Algorithm
transformation	I-Algorithm
on	O
.	O
</s>
<s>
It	O
also	O
restores	O
the	O
symmetry	O
between	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
and	O
its	O
inverse	O
.	O
</s>
<s>
Variations	O
of	O
all	O
three	O
conventions	O
can	O
be	O
created	O
by	O
conjugating	O
the	O
complex-exponential	O
kernel	B-Algorithm
of	O
both	O
the	O
forward	O
and	O
the	O
reverse	O
transform	B-Algorithm
.	O
</s>
<s>
As	O
discussed	O
above	O
,	O
the	O
characteristic	O
function	O
of	O
a	O
random	O
variable	O
is	O
the	O
same	O
as	O
the	O
Fourier	O
–	O
Stieltjes	O
transform	B-Algorithm
of	O
its	O
distribution	O
measure	O
,	O
but	O
in	O
this	O
context	O
it	O
is	O
typical	O
to	O
take	O
a	O
different	O
convention	O
for	O
the	O
constants	O
.	O
</s>
<s>
As	O
in	O
the	O
case	O
of	O
the	O
"	O
non-unitary	O
angular	O
frequency	O
"	O
convention	O
above	O
,	O
the	O
factor	O
of	O
2	O
appears	O
in	O
neither	O
the	O
normalizing	O
constant	O
nor	O
the	O
exponent	O
.	O
</s>
<s>
Since	O
the	O
fundamental	O
definition	O
of	O
a	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
an	O
integral	O
,	O
functions	O
that	O
can	O
be	O
expressed	O
as	O
closed-form	O
expressions	O
are	O
commonly	O
computed	O
by	O
working	O
the	O
integral	O
analytically	O
to	O
yield	O
a	O
closed-form	O
expression	O
in	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
conjugate	O
variable	O
as	O
the	O
result	O
.	O
</s>
<s>
This	O
is	O
the	O
method	O
used	O
to	O
generate	O
tables	O
of	O
Fourier	B-Algorithm
transforms	I-Algorithm
,	O
including	O
those	O
found	O
in	O
the	O
table	O
below	O
(	O
Fourier	B-Algorithm
transform	I-Algorithm
#Tables	O
of	O
important	O
Fourier	B-Algorithm
transforms	I-Algorithm
)	O
.	O
</s>
<s>
Many	O
computer	O
algebra	O
systems	O
such	O
as	O
Matlab	B-Language
and	O
Mathematica	B-Language
that	O
are	O
capable	O
of	O
symbolic	B-Algorithm
integration	I-Algorithm
are	O
capable	O
of	O
computing	O
Fourier	B-Algorithm
transforms	I-Algorithm
analytically	O
.	O
</s>
<s>
For	O
example	O
,	O
to	O
compute	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
one	O
might	O
enter	O
the	O
command	O
into	O
Wolfram	B-Application
Alpha	I-Application
.	O
</s>
<s>
If	O
the	O
input	O
function	O
is	O
in	O
closed-form	O
and	O
the	O
desired	O
output	O
function	O
is	O
a	O
series	O
of	O
ordered	O
pairs	O
(	O
for	O
example	O
a	O
table	O
of	O
values	O
from	O
which	O
a	O
graph	O
can	O
be	O
generated	O
)	O
over	O
a	O
specified	O
domain	B-Algorithm
,	O
then	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
can	O
be	O
generated	O
by	O
numerical	B-Algorithm
integration	I-Algorithm
at	O
each	O
value	O
of	O
the	O
Fourier	O
conjugate	O
variable	O
(	O
frequency	O
,	O
for	O
example	O
)	O
for	O
which	O
a	O
value	O
of	O
the	O
output	O
variable	O
is	O
desired	O
.	O
</s>
<s>
Note	O
that	O
this	O
method	O
requires	O
computing	O
a	O
separate	O
numerical	B-Algorithm
integration	I-Algorithm
for	O
each	O
value	O
of	O
frequency	O
for	O
which	O
a	O
value	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
desired	O
.	O
</s>
<s>
The	O
numerical	B-Algorithm
integration	I-Algorithm
approach	O
works	O
on	O
a	O
much	O
broader	O
class	O
of	O
functions	O
than	O
the	O
analytic	O
approach	O
,	O
because	O
it	O
yields	O
results	O
for	O
functions	O
that	O
do	O
not	O
have	O
closed	O
form	O
Fourier	B-Algorithm
transform	I-Algorithm
integrals	O
.	O
</s>
<s>
If	O
the	O
input	O
function	O
is	O
a	O
series	O
of	O
ordered	O
pairs	O
(	O
for	O
example	O
,	O
a	O
time	O
series	O
from	O
measuring	O
an	O
output	O
variable	O
repeatedly	O
over	O
a	O
time	O
interval	O
)	O
then	O
the	O
output	O
function	O
must	O
also	O
be	O
a	O
series	O
of	O
ordered	O
pairs	O
(	O
for	O
example	O
,	O
a	O
complex	O
number	O
vs.	O
frequency	O
over	O
a	O
specified	O
domain	B-Algorithm
of	O
frequencies	O
)	O
,	O
unless	O
certain	O
assumptions	O
and	O
approximations	O
are	O
made	O
allowing	O
the	O
output	O
function	O
to	O
be	O
approximated	O
by	O
a	O
closed-form	O
expression	O
.	O
</s>
<s>
In	O
the	O
general	O
case	O
where	O
the	O
available	O
input	O
series	O
of	O
ordered	O
pairs	O
are	O
assumed	O
be	O
samples	O
representing	O
a	O
continuous	O
function	O
over	O
an	O
interval	O
(	O
amplitude	B-Application
vs.	O
time	O
,	O
for	O
example	O
)	O
,	O
the	O
series	O
of	O
ordered	O
pairs	O
representing	O
the	O
desired	O
output	O
function	O
can	O
be	O
obtained	O
by	O
numerical	B-Algorithm
integration	I-Algorithm
of	O
the	O
input	O
data	O
over	O
the	O
available	O
interval	O
at	O
each	O
value	O
of	O
the	O
Fourier	O
conjugate	O
variable	O
(	O
frequency	O
,	O
for	O
example	O
)	O
for	O
which	O
the	O
value	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
desired	O
.	O
</s>
<s>
Explicit	O
numerical	B-Algorithm
integration	I-Algorithm
over	O
the	O
ordered	O
pairs	O
can	O
yield	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
output	O
value	O
for	O
any	O
desired	O
value	O
of	O
the	O
conjugate	O
Fourier	B-Algorithm
transform	I-Algorithm
variable	O
(	O
frequency	O
,	O
for	O
example	O
)	O
,	O
so	O
that	O
a	O
spectrum	O
can	O
be	O
produced	O
at	O
any	O
desired	O
step	O
size	O
and	O
over	O
any	O
desired	O
variable	O
range	B-Algorithm
for	O
accurate	O
determination	O
of	O
amplitudes	B-Application
,	O
frequencies	O
,	O
and	O
phases	O
corresponding	O
to	O
isolated	O
peaks	O
.	O
</s>
<s>
Unlike	O
limitations	O
in	O
DFT	O
and	O
FFT	O
methods	O
,	O
explicit	O
numerical	B-Algorithm
integration	I-Algorithm
can	O
have	O
any	O
desired	O
step	O
size	O
and	O
compute	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
over	O
any	O
desired	O
range	B-Algorithm
of	O
the	O
conjugate	O
Fourier	B-Algorithm
transform	I-Algorithm
variable	O
(	O
for	O
example	O
,	O
frequency	O
)	O
.	O
</s>
<s>
If	O
the	O
ordered	O
pairs	O
representing	O
the	O
original	O
input	O
function	O
are	O
equally	O
spaced	O
in	O
their	O
input	O
variable	O
(	O
for	O
example	O
,	O
equal	O
time	O
steps	O
)	O
,	O
then	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
known	O
as	O
a	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
(	O
DFT	O
)	O
,	O
which	O
can	O
be	O
computed	O
either	O
by	O
explicit	O
numerical	B-Algorithm
integration	I-Algorithm
,	O
by	O
explicit	O
evaluation	O
of	O
the	O
DFT	O
definition	O
,	O
or	O
by	O
fast	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
FFT	O
)	O
methods	O
.	O
</s>
<s>
In	O
contrast	O
to	O
explicit	O
integration	O
of	O
input	O
data	O
,	O
use	O
of	O
the	O
DFT	O
and	O
FFT	O
methods	O
produces	O
Fourier	B-Algorithm
transforms	I-Algorithm
described	O
by	O
ordered	O
pairs	O
of	O
step	O
size	O
equal	O
to	O
the	O
reciprocal	O
of	O
the	O
original	O
sampling	O
interval	O
.	O
</s>
<s>
The	O
following	O
tables	O
record	O
some	O
closed-form	O
Fourier	B-Algorithm
transforms	I-Algorithm
.	O
</s>
<s>
For	O
functions	O
and	O
denote	O
their	O
Fourier	B-Algorithm
transforms	I-Algorithm
by	O
and	O
.	O
</s>
<s>
It	O
may	O
be	O
useful	O
to	O
notice	O
that	O
entry	O
105	O
gives	O
a	O
relationship	O
between	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
a	O
function	O
and	O
the	O
original	O
function	O
,	O
which	O
can	O
be	O
seen	O
as	O
relating	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
and	O
its	O
inverse	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transforms	I-Algorithm
in	O
this	O
table	O
may	O
be	O
found	O
in	O
or	O
.	O
</s>
<s>
Function	O
Fourier	B-Algorithm
transform	I-Algorithm
unitary	B-Algorithm
,	O
ordinary	O
frequency	O
Fourier	B-Algorithm
transform	I-Algorithm
unitary	B-Algorithm
,	O
angular	O
frequency	O
Fourier	B-Algorithm
transform	I-Algorithm
non-unitary	O
,	O
angular	O
frequency	O
RemarksDefinitions	O
101Linearity	O
102Shift	O
in	O
time	O
domain	B-Algorithm
103Shift	O
in	O
frequency	O
domain	B-Algorithm
,	O
dual	O
of	O
102	O
104Scaling	O
in	O
the	O
time	O
domain	B-Algorithm
.	O
</s>
<s>
105The	O
same	O
transform	B-Algorithm
is	O
applied	O
twice	O
,	O
but	O
x	O
replaces	O
the	O
frequency	O
variable	O
( ξ	O
or	O
ω	B-General_Concept
)	O
after	O
the	O
first	O
transform	B-Algorithm
.	O
</s>
<s>
106as	O
is	O
a	O
Schwartz	B-Algorithm
function	I-Algorithm
107This	O
is	O
the	O
dual	O
of	O
106	O
108The	O
notation	O
denotes	O
the	O
convolution	B-Language
of	O
and	O
—	O
this	O
rule	O
is	O
the	O
convolution	B-Language
theorem	O
109This	O
is	O
the	O
dual	O
of	O
108	O
110For	O
purely	O
realHermitian	O
symmetry	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transforms	I-Algorithm
in	O
this	O
table	O
may	O
be	O
found	O
in	O
,	O
,	O
or	O
.	O
</s>
<s>
Function	O
Fourier	B-Algorithm
transform	I-Algorithm
unitary	B-Algorithm
,	O
ordinary	O
frequency	O
Fourier	B-Algorithm
transform	I-Algorithm
unitary	B-Algorithm
,	O
angular	O
frequency	O
Fourier	B-Algorithm
transform	I-Algorithm
non-unitary	O
,	O
angular	O
frequency	O
RemarksDefinitions	O
201The	O
rectangular	O
pulse	O
and	O
the	O
normalized	B-Algorithm
sinc	I-Algorithm
function	I-Algorithm
,	O
here	O
defined	O
as	O
202Dual	O
of	O
rule	O
201	O
.	O
</s>
<s>
The	O
rectangular	O
function	O
is	O
an	O
ideal	O
low-pass	B-Algorithm
filter	I-Algorithm
,	O
and	O
the	O
sinc	B-Algorithm
function	I-Algorithm
is	O
the	O
non-causal	O
impulse	O
response	O
of	O
such	O
a	O
filter	O
.	O
</s>
<s>
The	O
sinc	B-Algorithm
function	I-Algorithm
is	O
defined	O
here	O
as	O
203	O
The	O
function	O
is	O
the	O
triangular	O
function	O
204	O
Dual	O
of	O
rule	O
203	O
.	O
</s>
<s>
206This	O
shows	O
that	O
,	O
for	O
the	O
unitary	B-Algorithm
Fourier	B-Algorithm
transforms	I-Algorithm
,	O
the	O
Gaussian	O
function	O
is	O
its	O
own	O
Fourier	B-Algorithm
transform	I-Algorithm
for	O
some	O
choice	O
of	O
.	O
</s>
<s>
That	O
is	O
,	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
a	O
two-sided	O
decaying	O
exponential	O
function	O
is	O
a	O
Lorentzian	O
function	O
.	O
</s>
<s>
209Hyperbolic	O
secant	O
is	O
its	O
own	O
Fourier	B-Algorithm
transform	I-Algorithm
210	O
is	O
the	O
th-order	O
Hermite	O
polynomial	O
.	O
</s>
<s>
If	O
then	O
the	O
Gauss	O
–	O
Hermite	O
functions	O
are	O
eigenfunctions	B-Algorithm
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
operator	O
.	O
</s>
<s>
For	O
a	O
derivation	B-Algorithm
,	O
see	O
Hermite	O
polynomial	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transforms	I-Algorithm
in	O
this	O
table	O
may	O
be	O
found	O
in	O
or	O
.	O
</s>
<s>
Function	O
Fourier	B-Algorithm
transform	I-Algorithm
unitary	B-Algorithm
,	O
ordinary	O
frequency	O
Fourier	B-Algorithm
transform	I-Algorithm
unitary	B-Algorithm
,	O
angular	O
frequency	O
Fourier	B-Algorithm
transform	I-Algorithm
non-unitary	O
,	O
angular	O
frequency	O
RemarksDefinitions	O
301The	O
distribution	O
denotes	O
the	O
Dirac	O
delta	O
function	O
.	O
</s>
<s>
309Here	O
,	O
is	O
a	O
natural	O
number	O
and	O
is	O
the	O
th	O
distribution	O
derivative	B-Algorithm
of	O
the	O
Dirac	O
delta	O
function	O
.	O
</s>
<s>
Combining	O
this	O
rule	O
with	O
101	O
,	O
we	O
can	O
transform	B-Algorithm
all	O
polynomials	O
.	O
</s>
<s>
310Dual	O
of	O
rule	O
309	O
.	O
is	O
the	O
th	O
distribution	O
derivative	B-Algorithm
of	O
the	O
Dirac	O
delta	O
function	O
.	O
</s>
<s>
It	O
is	O
necessary	O
to	O
use	O
the	O
Cauchy	O
principal	O
value	O
when	O
testing	O
against	O
Schwartz	B-Algorithm
functions	I-Algorithm
.	O
</s>
<s>
This	O
rule	O
is	O
useful	O
in	O
studying	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
312	O
is	O
the	O
homogeneous	O
distribution	O
defined	O
by	O
the	O
distributional	O
derivative	B-Algorithm
313This	O
formula	O
is	O
valid	O
for	O
.	O
</s>
<s>
This	O
time	O
the	O
Fourier	B-Algorithm
transforms	I-Algorithm
need	O
to	O
be	O
considered	O
as	O
a	O
Cauchy	O
principal	O
value	O
.	O
</s>
<s>
It	O
is	O
necessary	O
to	O
use	O
a	O
finite	O
part	O
integral	O
when	O
testing	O
or	O
against	O
Schwartz	B-Algorithm
functions	I-Algorithm
.	O
</s>
<s>
Use	O
differentiation	B-Algorithm
to	O
derive	O
formula	O
for	O
higher	O
exponents	O
.	O
</s>
<s>
Function	O
Fourier	B-Algorithm
transform	I-Algorithm
unitary	B-Algorithm
,	O
ordinary	O
frequency	O
Fourier	B-Algorithm
transform	I-Algorithm
unitary	B-Algorithm
,	O
angular	O
frequency	O
Fourier	B-Algorithm
transform	I-Algorithm
non-unitary	O
,	O
angular	O
frequency	O
Remarks400The	O
variables	O
,	O
,	O
,	O
are	O
real	O
numbers	O
.	O
</s>
<s>
Function	O
Fourier	B-Algorithm
transform	I-Algorithm
unitary	B-Algorithm
,	O
ordinary	O
frequency	O
Fourier	B-Algorithm
transform	I-Algorithm
unitary	B-Algorithm
,	O
angular	O
frequency	O
Fourier	B-Algorithm
transform	I-Algorithm
non-unitary	O
,	O
angular	O
frequency	O
Remarks500501The	O
function	O
is	O
the	O
indicator	O
function	O
of	O
the	O
interval	O
.	O
</s>
<s>
Taking	O
and	O
produces	O
402	O
..	O
502See	O
Riesz	O
potential	O
where	O
the	O
constant	O
is	O
given	O
byThe	O
formula	O
also	O
holds	O
for	O
all	O
by	O
analytic	O
continuation	O
,	O
but	O
then	O
the	O
function	O
and	O
its	O
Fourier	B-Algorithm
transforms	I-Algorithm
need	O
to	O
be	O
understood	O
as	O
suitably	O
regularized	O
tempered	O
distributions	O
.	O
</s>
<s>
See	O
homogeneous	O
distribution.In	O
,	O
with	O
the	O
non-unitary	O
conventions	O
of	O
this	O
table	O
,	O
the	O
transform	B-Algorithm
of	O
is	O
given	O
to	O
be	O
from	O
which	O
this	O
follows	O
,	O
with	O
.503This	O
is	O
the	O
formula	O
for	O
a	O
multivariate	O
normal	O
distribution	O
normalized	O
to	O
1	O
with	O
a	O
mean	O
of	O
0	O
.	O
</s>
