Digital Signal Processing Reference
In-Depth Information
This expression is of the same kind as the customary formula encountered
in the wellknown Gauss numerical integrations with the help of the classical
polynomials. The only difference is that we employ the nonclassical Lanczos
polynomials in (2.74). Naturally, in all realistic computations, the infinite up
per limit in the sum over n from (2.74) is replaced by a finite cutoff number
K which is the order or rank of the quadrature rule. Thus, when the develop
ment in (2.74) is truncated to the first K terms, the following approximation
is obtained
≈
K
b
dσ(x)
z−x
w
k
z−x
k
.
(2.75)
a
k=1
Importantly, even with only K terms retained in the Heaviside partial frac
tions, the final result for S(z) can still be exact if the errorE
K
(z) is taken
into account via
K
b
dσ(x)
z−x
=
w
k
z−x
k
S(z)≡
+E
K
(z)
a
k=1
=
0
β
1
P
K
(z)
Q
K
(z)
+E
K
(z).
(2.76)
This analysis can be extended to the corresponding exact Gausstype quadra
ture for the case of a general function f(x) in place of (z−x)
−1
n
b
f(x)dσ(x) = lim
n→∞
w
k
f(x
k
) +E
n
(2.77)
a
k=1
b
1
Q
n
(z)
E
n
=
dσ(x)f(x)Q
n
(x).
(2.78)
a
∞
Employing the binomial expansion for (z−x)
−1
n=0
x
n
z
−n−1
=
in (2.67),
we have
∞
n,m
z
−m−1
π
n
(z) =
(2.79)
m=0
where
n,m
is the modified moment [5]
b
n,m
=Q
n
(x)|x
m
=
Q
∗
n
(x)x
m
dσ(x)
n,0
=
n
.
(2.80)
a
By means of the orthogonality (2.34), the following important feature of
n,m
is deduced
Q
n
(x)|x
m
= 0
m = 0, 1,...,n−1
∴
n,m
= 0
m≤n−1.
(2.81)
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