<s>
In	O
mathematics	O
,	O
the	O
Hankel	B-Algorithm
transform	I-Algorithm
expresses	O
any	O
given	O
function	O
f(r )	O
as	O
the	O
weighted	O
sum	O
of	O
an	O
infinite	O
number	O
of	O
Bessel	O
functions	O
of	O
the	O
first	O
kind	O
.	O
</s>
<s>
The	O
Hankel	B-Algorithm
transform	I-Algorithm
is	O
an	O
integral	B-Algorithm
transform	I-Algorithm
and	O
was	O
first	O
developed	O
by	O
the	O
mathematician	O
Hermann	O
Hankel	O
.	O
</s>
<s>
It	O
is	O
also	O
known	O
as	O
the	O
Fourier	B-Algorithm
–	I-Algorithm
Bessel	I-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
Just	O
as	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
for	O
an	O
infinite	O
interval	O
is	O
related	O
to	O
the	O
Fourier	O
series	O
over	O
a	O
finite	O
interval	O
,	O
so	O
the	O
Hankel	B-Algorithm
transform	I-Algorithm
over	O
an	O
infinite	O
interval	O
is	O
related	O
to	O
the	O
Fourier	O
–	O
Bessel	O
series	O
over	O
a	O
finite	O
interval	O
.	O
</s>
<s>
However	O
,	O
like	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
the	O
domain	O
can	O
be	O
extended	O
by	O
a	O
density	O
argument	O
to	O
include	O
some	O
functions	O
whose	O
above	O
integral	O
is	O
not	O
finite	O
,	O
for	O
example	O
.	O
</s>
<s>
This	O
means	O
that	O
,	O
as	O
with	O
the	O
previous	O
definition	O
,	O
the	O
Hankel	B-Algorithm
transform	I-Algorithm
defined	O
this	O
way	O
is	O
also	O
its	O
own	O
inverse	O
:	O
</s>
<s>
According	O
to	O
the	O
reference	O
given	O
above	O
,	O
we	O
can	O
take	O
the	O
integral	O
as	O
the	O
limit	O
as	O
the	O
upper	O
limit	O
goes	O
to	O
infinity	O
(	O
an	O
improper	O
integral	O
rather	O
than	O
a	O
Lebesgue	O
integral	O
)	O
,	O
and	O
in	O
this	O
way	O
the	O
Hankel	B-Algorithm
transform	I-Algorithm
and	O
its	O
inverse	O
work	O
for	O
all	O
functions	O
in	O
L2( 0	O
,	O
∞	O
)	O
.	O
</s>
<s>
The	O
Hankel	B-Algorithm
transform	I-Algorithm
can	O
be	O
used	O
to	O
transform	O
and	O
solve	O
Laplace	O
's	O
equation	O
expressed	O
in	O
cylindrical	O
coordinates	O
.	O
</s>
<s>
Under	O
the	O
Hankel	B-Algorithm
transform	I-Algorithm
,	O
the	O
Bessel	O
operator	O
becomes	O
a	O
multiplication	O
by	O
.	O
</s>
<s>
The	O
Bessel	O
functions	O
form	O
an	O
orthogonal	B-Algorithm
basis	I-Algorithm
with	O
respect	O
to	O
the	O
weighting	O
factor	O
r	O
:	O
</s>
<s>
The	O
Hankel	B-Algorithm
transform	I-Algorithm
appears	O
when	O
one	O
writes	O
the	O
multidimensional	O
Fourier	B-Algorithm
transform	I-Algorithm
in	O
hyperspherical	O
coordinates	O
,	O
which	O
is	O
the	O
reason	O
why	O
the	O
Hankel	B-Algorithm
transform	I-Algorithm
often	O
appears	O
in	O
physical	O
problems	O
with	O
cylindrical	O
or	O
spherical	O
symmetry	O
.	O
</s>
<s>
Its	O
-dimensional	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
defined	O
asTo	O
rewrite	O
it	O
in	O
hyperspherical	O
coordinates	O
,	O
we	O
can	O
use	O
the	O
decomposition	O
of	O
a	O
plane	O
wave	O
into	O
-dimensional	O
hyperspherical	O
harmonics	O
:where	O
and	O
are	O
the	O
sets	O
of	O
all	O
hyperspherical	O
angles	O
in	O
the	O
-space	O
and	O
-space	O
.	O
</s>
<s>
This	O
gives	O
the	O
following	O
expression	O
for	O
the	O
-dimensional	O
Fourier	B-Algorithm
transform	I-Algorithm
in	O
hyperspherical	O
coordinates:If	O
we	O
expand	O
and	O
in	O
hyperspherical	O
harmonics:the	O
Fourier	B-Algorithm
transform	I-Algorithm
in	O
hyperspherical	O
coordinates	O
simplifies	O
toThis	O
means	O
that	O
functions	O
with	O
angular	O
dependence	O
in	O
form	O
of	O
a	O
hyperspherical	O
harmonic	O
retain	O
it	O
upon	O
the	O
multidimensional	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
while	O
the	O
radial	O
part	O
undergoes	O
the	O
Hankel	B-Algorithm
transform	I-Algorithm
(	O
up	O
to	O
some	O
extra	O
factors	O
like	O
)	O
.	O
</s>
<s>
then	O
its	O
two-dimensional	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
given	O
bywhereis	O
the	O
-th	O
order	O
Hankel	B-Algorithm
transform	I-Algorithm
of	O
(	O
in	O
this	O
case	O
plays	O
the	O
role	O
of	O
the	O
angular	O
momentum	O
,	O
which	O
was	O
denoted	O
by	O
in	O
the	O
previous	O
section	O
)	O
.	O
</s>
<s>
then	O
its	O
three-dimensional	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
given	O
bywhereis	O
the	O
Hankel	B-Algorithm
transform	I-Algorithm
of	O
of	O
order	O
.	O
</s>
<s>
This	O
kind	O
of	O
Hankel	B-Algorithm
transform	I-Algorithm
of	O
half-integer	O
order	O
is	O
also	O
known	O
as	O
the	O
spherical	O
Bessel	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
If	O
a	O
-dimensional	O
function	O
does	O
not	O
depend	O
on	O
angular	O
coordinates	O
,	O
then	O
its	O
-dimensional	O
Fourier	B-Algorithm
transform	I-Algorithm
also	O
does	O
not	O
depend	O
on	O
angular	O
coordinates	O
and	O
is	O
given	O
bywhich	O
is	O
the	O
Hankel	B-Algorithm
transform	I-Algorithm
of	O
of	O
order	O
up	O
to	O
a	O
factor	O
of	O
.	O
</s>
<s>
This	O
is	O
one	O
flavor	O
of	O
fast	O
Hankel	B-Algorithm
transform	I-Algorithm
techniques	O
.	O
</s>
<s>
The	O
Hankel	B-Algorithm
transform	I-Algorithm
is	O
one	O
member	O
of	O
the	O
FHA	B-Algorithm
cycle	I-Algorithm
of	O
integral	B-Algorithm
operators	I-Algorithm
.	O
</s>
<s>
In	O
other	O
words	O
,	O
applying	O
the	O
Abel	B-Algorithm
transform	I-Algorithm
to	O
a	O
1-dimensional	O
function	O
and	O
then	O
applying	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
to	O
that	O
result	O
is	O
the	O
same	O
as	O
applying	O
the	O
Hankel	B-Algorithm
transform	I-Algorithm
to	O
that	O
function	O
.	O
</s>
<s>
Now	O
the	O
integral	O
can	O
be	O
calculated	O
numerically	O
with	O
complexity	O
using	O
fast	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
The	O
algorithm	O
can	O
be	O
further	O
simplified	O
by	O
using	O
a	O
known	O
analytical	O
expression	O
for	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
:	O
</s>
<s>
This	O
algorithm	O
is	O
known	O
as	O
the	O
"	O
quasi-fast	O
Hankel	B-Algorithm
transform	I-Algorithm
"	O
,	O
or	O
simply	O
"	O
fast	O
Hankel	B-Algorithm
transform	I-Algorithm
"	O
.	O
</s>
<s>
Since	O
it	O
is	O
based	O
on	O
fast	O
Fourier	B-Algorithm
transform	I-Algorithm
in	O
logarithmic	O
variables	O
,	O
has	O
to	O
be	O
defined	O
on	O
a	O
logarithmic	O
grid	O
.	O
</s>
<s>
For	O
functions	O
defined	O
on	O
a	O
uniform	O
grid	O
,	O
a	O
number	O
of	O
other	O
algorithms	O
exist	O
,	O
including	O
straightforward	O
quadrature	B-Algorithm
,	O
methods	O
based	O
on	O
the	O
projection-slice	B-Algorithm
theorem	I-Algorithm
,	O
and	O
methods	O
using	O
the	O
asymptotic	O
expansion	O
of	O
Bessel	O
functions	O
.	O
</s>
<s>
The	O
Hankel	B-Algorithm
transform	I-Algorithm
of	O
Zernike	O
polynomials	O
are	O
essentially	O
Bessel	O
Functions	O
(	O
Noll	O
1976	O
)	O
:	O
</s>
