<s>
In	O
mathematics	O
,	O
physics	O
and	O
engineering	O
,	O
the	O
sinc	B-Algorithm
function	I-Algorithm
,	O
denoted	O
by	O
,	O
has	O
two	O
forms	O
,	O
normalized	O
and	O
unnormalized	O
.	O
</s>
<s>
Alternatively	O
,	O
the	O
unnormalized	B-Algorithm
sinc	I-Algorithm
function	I-Algorithm
is	O
often	O
called	O
the	O
sampling	O
function	O
,	O
indicated	O
as	O
Sa(x )	O
.	O
</s>
<s>
The	O
normalization	O
causes	O
the	O
definite	O
integral	O
of	O
the	O
function	O
over	O
the	O
real	O
numbers	O
to	O
equal	O
1	O
(	O
whereas	O
the	O
same	O
integral	O
of	O
the	O
unnormalized	B-Algorithm
sinc	I-Algorithm
function	I-Algorithm
has	O
a	O
value	O
of	O
)	O
.	O
</s>
<s>
As	O
a	O
further	O
useful	O
property	O
,	O
the	O
zeros	O
of	O
the	O
normalized	B-Algorithm
sinc	I-Algorithm
function	I-Algorithm
are	O
the	O
nonzero	O
integer	O
values	O
of	O
.	O
</s>
<s>
The	O
normalized	B-Algorithm
sinc	I-Algorithm
function	I-Algorithm
is	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
the	O
rectangular	O
function	O
with	O
no	O
scaling	O
.	O
</s>
<s>
The	O
sinc	B-Algorithm
function	I-Algorithm
is	O
then	O
analytic	B-Language
everywhere	O
and	O
hence	O
an	O
entire	O
function	O
.	O
</s>
<s>
The	O
term	O
sinc	B-Algorithm
was	O
introduced	O
by	O
Philip	O
M	O
.	O
Woodward	O
in	O
his	O
1952	O
article	O
"	O
Information	O
theory	O
and	O
inverse	O
probability	O
in	O
telecommunication	O
"	O
,	O
in	O
which	O
he	O
said	O
that	O
the	O
function	O
"	O
occurs	O
so	O
often	O
in	O
Fourier	O
analysis	O
and	O
its	O
applications	O
that	O
it	O
does	O
seem	O
to	O
merit	O
some	O
notation	O
of	O
its	O
own	O
"	O
,	O
and	O
his	O
1953	O
book	O
Probability	O
and	O
Information	O
Theory	O
,	O
with	O
Applications	O
to	O
Radar	O
.	O
</s>
<s>
The	O
zero	B-Algorithm
crossings	I-Algorithm
of	O
the	O
unnormalized	O
sinc	B-Algorithm
are	O
at	O
non-zero	O
integer	O
multiples	O
of	O
,	O
while	O
zero	B-Algorithm
crossings	I-Algorithm
of	O
the	O
normalized	O
sinc	B-Algorithm
occur	O
at	O
non-zero	O
integers	O
.	O
</s>
<s>
The	O
local	O
maxima	O
and	O
minima	O
of	O
the	O
unnormalized	O
sinc	B-Algorithm
correspond	O
to	O
its	O
intersections	O
with	O
the	O
cosine	O
function	O
.	O
</s>
<s>
This	O
follows	O
from	O
the	O
derivative	O
of	O
the	O
sinc	B-Algorithm
function	I-Algorithm
:	O
</s>
<s>
The	O
normalized	B-Algorithm
sinc	I-Algorithm
function	I-Algorithm
has	O
a	O
simple	O
representation	O
as	O
the	O
infinite	O
product	O
:	O
</s>
<s>
and	O
is	O
related	O
to	O
the	O
gamma	O
function	O
through	O
Euler	B-Language
's	O
reflection	O
formula	O
:	O
</s>
<s>
The	O
continuous	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
of	O
the	O
normalized	O
sinc	B-Algorithm
(	O
to	O
ordinary	O
frequency	O
)	O
is	O
:	O
</s>
<s>
This	O
corresponds	O
to	O
the	O
fact	O
that	O
the	O
sinc	B-Algorithm
filter	I-Algorithm
is	O
the	O
ideal	O
(	O
brick-wall	O
,	O
meaning	O
rectangular	O
frequency	O
response	O
)	O
low-pass	B-Algorithm
filter	I-Algorithm
.	O
</s>
<s>
The	O
normalized	B-Algorithm
sinc	I-Algorithm
function	I-Algorithm
has	O
properties	O
that	O
make	O
it	O
ideal	O
in	O
relationship	O
to	O
interpolation	B-Algorithm
of	O
sampled	B-Algorithm
bandlimited	O
functions	O
:	O
</s>
<s>
Other	O
properties	O
of	O
the	O
two	O
sinc	B-Algorithm
functions	I-Algorithm
include	O
:	O
</s>
<s>
The	O
unnormalized	O
sinc	B-Algorithm
is	O
the	O
zeroth-order	O
spherical	O
Bessel	O
function	O
of	O
the	O
first	O
kind	O
,	O
.	O
</s>
<s>
The	O
normalized	O
sinc	B-Algorithm
is	O
.	O
</s>
<s>
(	O
not	O
normalized	O
)	O
is	O
one	O
of	O
two	O
linearly	O
independent	O
solutions	O
to	O
the	O
linear	O
ordinary	O
differential	O
equation	O
The	O
other	O
is	O
,	O
which	O
is	O
not	O
bounded	O
at	O
,	O
unlike	O
its	O
sinc	B-Algorithm
function	I-Algorithm
counterpart	O
.	O
</s>
<s>
Using	O
normalized	O
sinc	B-Algorithm
,	O
</s>
<s>
The	O
following	O
improper	O
integral	O
involves	O
the	O
(	O
not	O
normalized	O
)	O
sinc	B-Algorithm
function	I-Algorithm
:	O
</s>
<s>
The	O
normalized	B-Algorithm
sinc	I-Algorithm
function	I-Algorithm
can	O
be	O
used	O
as	O
a	O
nascent	O
delta	O
function	O
,	O
meaning	O
that	O
the	O
following	O
weak	B-Algorithm
limit	I-Algorithm
holds	O
:	O
</s>
<s>
for	O
every	O
Schwartz	B-Algorithm
function	I-Algorithm
,	O
as	O
can	O
be	O
seen	O
from	O
the	O
Fourier	O
inversion	O
theorem	O
.	O
</s>
<s>
In	O
the	O
above	O
expression	O
,	O
as	O
,	O
the	O
number	O
of	O
oscillations	O
per	O
unit	O
length	O
of	O
the	O
sinc	B-Algorithm
function	I-Algorithm
approaches	O
infinity	O
.	O
</s>
<s>
All	O
sums	O
in	O
this	O
section	O
refer	O
to	O
the	O
unnormalized	B-Algorithm
sinc	I-Algorithm
function	I-Algorithm
.	O
</s>
<s>
Euler	B-Language
famously	O
compared	O
this	O
series	O
to	O
the	O
expansion	O
of	O
the	O
infinite	O
product	O
form	O
to	O
solve	O
the	O
Basel	O
problem	O
.	O
</s>
<s>
The	O
product	O
of	O
1-D	O
sinc	B-Algorithm
functions	I-Algorithm
readily	O
provides	O
a	O
multivariate	O
sinc	B-Algorithm
function	I-Algorithm
for	O
the	O
square	O
Cartesian	O
grid	O
(	O
lattice	O
)	O
:	O
,	O
whose	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
the	O
indicator	O
function	O
of	O
a	O
square	O
in	O
the	O
frequency	O
space	O
(	O
i.e.	O
,	O
the	O
brick	O
wall	O
defined	O
in	O
2-D	O
space	O
)	O
.	O
</s>
<s>
The	O
sinc	B-Algorithm
function	I-Algorithm
for	O
a	O
non-Cartesian	O
lattice	O
(	O
e.g.	O
,	O
hexagonal	O
lattice	O
)	O
is	O
a	O
function	O
whose	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
the	O
indicator	O
function	O
of	O
the	O
Brillouin	O
zone	O
of	O
that	O
lattice	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
sinc	B-Algorithm
function	I-Algorithm
for	O
the	O
hexagonal	O
lattice	O
is	O
a	O
function	O
whose	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
the	O
indicator	O
function	O
of	O
the	O
unit	O
hexagon	O
in	O
the	O
frequency	O
space	O
.	O
</s>
<s>
However	O
,	O
the	O
explicit	O
formula	O
for	O
the	O
sinc	B-Algorithm
function	I-Algorithm
for	O
the	O
hexagonal	O
,	O
body-centered	O
cubic	O
,	O
face-centered	O
cubic	O
and	O
other	O
higher-dimensional	O
lattices	O
can	O
be	O
explicitly	O
derived	O
using	O
the	O
geometric	O
properties	O
of	O
Brillouin	O
zones	O
and	O
their	O
connection	O
to	O
zonotopes	O
.	O
</s>
