<s>
In	O
linear	B-Language
algebra	I-Language
,	O
a	O
Hankel	B-Algorithm
matrix	I-Algorithm
(	O
or	O
catalecticant	O
matrix	O
)	O
,	O
named	O
after	O
Hermann	O
Hankel	O
,	O
is	O
a	O
square	B-Algorithm
matrix	I-Algorithm
in	O
which	O
each	O
ascending	O
skew-diagonal	O
from	O
left	O
to	O
right	O
is	O
constant	O
,	O
e.g.	O
</s>
<s>
The	O
Hankel	B-Algorithm
matrix	I-Algorithm
is	O
a	O
symmetric	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Let	O
be	O
the	O
exchange	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
Hankel	B-Algorithm
matrix	I-Algorithm
,	O
then	O
where	O
is	O
a	O
Toeplitz	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
If	O
is	O
real	O
symmetric	B-Algorithm
,	O
then	O
will	O
have	O
the	O
same	O
eigenvalues	O
as	O
up	O
to	O
sign	O
.	O
</s>
<s>
The	O
Hilbert	O
matrix	O
is	O
an	O
example	O
of	O
a	O
Hankel	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
A	O
Hankel	B-Algorithm
operator	I-Algorithm
on	O
a	O
Hilbert	O
space	O
is	O
one	O
whose	O
matrix	O
is	O
a	O
(	O
possibly	O
infinite	O
)	O
Hankel	B-Algorithm
matrix	I-Algorithm
with	O
respect	O
to	O
an	O
orthonormal	O
basis	O
.	O
</s>
<s>
As	O
indicated	O
above	O
,	O
a	O
Hankel	B-Algorithm
Matrix	I-Algorithm
is	O
a	O
matrix	O
with	O
constant	O
values	O
along	O
its	O
antidiagonals	O
,	O
which	O
means	O
that	O
a	O
Hankel	B-Algorithm
matrix	I-Algorithm
must	O
satisfy	O
,	O
for	O
all	O
rows	O
and	O
columns	O
,	O
.	O
</s>
<s>
Let	O
the	O
corresponding	O
Hankel	B-Algorithm
Operator	I-Algorithm
be	O
.	O
</s>
<s>
Given	O
a	O
Hankel	B-Algorithm
matrix	I-Algorithm
,	O
the	O
corresponding	O
Hankel	B-Algorithm
operator	I-Algorithm
is	O
then	O
defined	O
as	O
.	O
</s>
<s>
We	O
are	O
often	O
interested	O
in	O
Hankel	B-Algorithm
operators	I-Algorithm
over	O
the	O
Hilbert	O
space	O
,	O
the	O
space	O
of	O
square	O
integrable	O
bilateral	O
complex	O
sequences	O
.	O
</s>
<s>
We	O
are	O
often	O
interested	O
in	O
approximations	O
of	O
the	O
Hankel	B-Algorithm
operators	I-Algorithm
,	O
possibly	O
by	O
low-order	O
operators	O
.	O
</s>
<s>
We	O
also	O
require	O
that	O
the	O
approximation	O
is	O
a	O
Hankel	B-Algorithm
matrix	I-Algorithm
,	O
which	O
can	O
be	O
shown	O
with	O
AAK	O
theory	O
.	O
</s>
<s>
The	O
determinant	O
of	O
a	O
Hankel	B-Algorithm
matrix	I-Algorithm
is	O
called	O
a	O
catalecticant	O
.	O
</s>
<s>
The	O
Hankel	B-Algorithm
matrix	I-Algorithm
transform	O
,	O
or	O
simply	O
Hankel	O
transform	O
,	O
produces	O
the	O
sequence	O
of	O
the	O
determinants	O
of	O
the	O
Hankel	B-Algorithm
matrices	I-Algorithm
formed	O
from	O
the	O
given	O
sequence	O
.	O
</s>
<s>
Hankel	B-Algorithm
matrices	I-Algorithm
are	O
formed	O
when	O
,	O
given	O
a	O
sequence	O
of	O
output	O
data	O
,	O
a	O
realization	O
of	O
an	O
underlying	O
state-space	O
or	O
hidden	O
Markov	O
model	O
is	O
desired	O
.	O
</s>
<s>
The	O
singular	O
value	O
decomposition	O
of	O
the	O
Hankel	B-Algorithm
matrix	I-Algorithm
provides	O
a	O
means	O
of	O
computing	O
the	O
A	O
,	O
B	O
,	O
and	O
C	O
matrices	O
which	O
define	O
the	O
state-space	O
realization	O
.	O
</s>
<s>
The	O
Hankel	B-Algorithm
matrix	I-Algorithm
formed	O
from	O
the	O
signal	O
has	O
been	O
found	O
useful	O
for	O
decomposition	O
of	O
non-stationary	O
signals	O
and	O
time-frequency	O
representation	O
.	O
</s>
<s>
The	O
method	O
of	O
moments	O
applied	O
to	O
polynomial	O
distributions	O
results	O
in	O
a	O
Hankel	B-Algorithm
matrix	I-Algorithm
that	O
needs	O
to	O
be	O
inverted	O
in	O
order	O
to	O
obtain	O
the	O
weight	O
parameters	O
of	O
the	O
polynomial	O
distribution	O
approximation	O
.	O
</s>
