Digital Signal Processing Reference
In-Depth Information
d
(r,s)
k
=Φ
r
|Υ
k
Υ
k
|Φ
s
.
(2.48)
A simplification occurs in the diagonal case when the residue (2.48) is reduced
to d
(s)
k
≡d
(s
k
. We see from (2.47) that G
r,s
(z) is
the ratio of two polynomials A
K−1
(z)/B
K
(z), which is the Pade approximant,
as in (2.25), and that was set to prove (QED). This indicates that the Pade
approximant is the most natural direct method for obtaining the Green func
tion. Namely, by construction, the Pade approximant is exact whenever the
function to be modeled is itself a quotient of two polynomials. This is the case
with the Green function (2.47), which is a polynomial quotient, since it origi
nates from the matrix element of the operator Pade approximant represented
by the resolvent operator (2.40).
Due to their orthonormality, the polynomials{Q
r
(x)}fulfill the standard
threeterm recursion relation
where d
(s,s)
k
=|Φ
s
|Υ
k
|
2
β
n+1
Q
n+1
(x) = (x−α
n
)Q
n
(x)−β
n
Q
n−1
(x)
Q
−1
(x) = 0
.
(2.49)
Q
0
(x) = 1
When (2.49) is multiplied by Q
∗
m
(x) and the product integrated over x from
a to b with the help of (2.33) and (2.35), the following result is obtained
b
1
w
n
Q
∗
n
(x)xQ
n
(x)dσ(x).
α
n
=
(2.50)
a
Similarly, multiplication of (2.49) by Q
n−1
(x) yields the expression
b
1
w
n−1
Q
∗
n−1
(x)xQ
n
(x)dσ(x).
β
n
=
(2.51)
a
The formulae (2.50) and (2.51) can be taken as the definitions of the parame
ters α
n
and β
n
. To check these relations, we first eliminate the term xQ
n
(x)
from (2.50) and (2.51). Afterwards, using (2.35), (2.50) and (2.51) the iden
tities α
n
≡β
n
are obtained (QED). Hence, α
n
from (2.50)
represents the diagonal element of the position operator x in the coordinate
representation. Similarly, in this latter representation, the same operator x
contains the parameter β
n
from (2.51) as the coupling constant between the
n th and (n−1) st polynomials. We can arrive at an alternative interpreta
tion of β
n
. To this end, we initially eliminate xQ
∗
n−1
(x) from (2.51) employing
(2.49) so that xQ
∗
n−1
(x) = β
n
Q
∗
n
(x) + α
n−1
Q
∗
n−1
(x) + β
n−1
Q
∗
n−2
(x). This is
followed by the application of the orthonormality condition (2.35) with the
final result
≡α
n
and β
n
b
b
1
w
n−1
1
w
n−1
|Q
n
(x)|
2
dσ(x) =
Q
n
(x)dσ(x)
β
n
=
a
a
1
w
n−1
2
=⇒β
n
> 0.
=
Q
n
(2.52)
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