Digital Signal Processing Reference
In-Depth Information
Because of the scaled initial conditions relative to (2.49), the degree of the
polynomial P
n
(x) is by 1 smaller than that of Q
n
(x). Thus, the degrees of
the polynomials Q
n
(x) and P
n
(x) are n and n−1, respectively, as per (2.55).
There is a relationship between the polynomials Q
n
(x) and P
n
(x) by means
of the following integral
b
P
n
(z) =
β
1
0
Q
n
(z)−Q
n
(x)
z−x
dσ(x)
(2.59)
a
P
n
(z
∗
) = P
∗
n
(z)
(2.60)
where
0
is the simplest case (n = 0) of the power moment
b
x
n
dσ(x).
n
=
(2.61)
a
These polynomials can be linked to the standard Gaussian quadratures. To
this end, we introduce the Stieltjes integral S(z) via
b
dσ(x)
z−x
S(z
∗
) = S
∗
(z).
S(z) =
(2.62)
a
As usual, the Stieltjes integral can be represented by the Pade approximant
R
n
(u) via
S(z)≈R
n
(z)
(2.63)
n
(x)≡
0
β
1
P
n
(x)
Q
n
(x)
.
R
(2.64)
To estimate the remainder S(z)−R
n
(z) in (2.63), we define the error term
E
n
(z), which is a measure of the approximation S(z)≈R
n
(z)
S(z) =R
n
(z) +E
n
(z).
(2.65)
It is possible to derive an expression for the error or remainderE
n
(z) by
substituting (2.65) into (2.59) and employing (2.64)
b
b
P
n
(z) =
β
1
0
dσ(x)
z−x
dσ(x)
z−x
Q
n
(x)
Q
n
(z)
−
a
a
b
=
β
1
0
dσ(x)
z−x
Q
n
(x)
Q
n
(z)[R
n
(z) +E
n
(z)]−
a
b
=
β
1
0
0
β
1
P
n
(z)
Q
n
(z)
dσ(x)
z−x
Q
n
(x)
Q
n
(z)
+E
n
(z)
−
a
b
P
n
(z) = P
n
(z) +
β
1
0
dσ(x)
z−x
Q
n
(x)
Q
n
(z)E
n
(z)−
a
n
(z) =
π
n
(z)
Q
n
(z)
∴
E
(2.66)
b
dσ(x)
z−x
Q
n
(x).
π
n
(z)≡
(2.67)
a
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