Digital Signal Processing Reference
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where the weight function W(x) > 0 has at least n + 1 points of increase.
Next, in the space of polynomials{Q
r
(x)}, one can define the asymmetric
scalar/inner product with respect to the measure dσ(x) via
b
Q
∗
m
(x)Q
n
(x)dσ(x).
Q
m
(x)|Q
n
(x)≡
(2.34)
a
Of course, asymmetric scalar products are customary in the conventional for
mulation of quantum mechanics defined on the field of Hermitean operators.
Clearly, the star in Q
∗
m
(x) is superfluous in (2.34) because Q
∗
m
(x) = Q
m
(x)
according to (2.32). We shall keep the star superscript on all the 'bra' vectors
f(x)|even for a real function f(x) in order to indicate the use of the asymmet
ric inner product. For the value Q
n
(z) with a complex z, it is obvious that the
star superscript on such a polynomial cannot be ignored. Such circumstances
can be relevant for resolvents of even Hermitean operators (Hamiltonians)
encountered in quantum mechanics. Further, we assume that polynomials
{Q
n
(x)}are orthonormalized with respect to the measure dσ(x)
b
Q
∗
n
(x)Q
m
(x)dσ(x) = w
n
δ
nm
(2.35)
a
where the coe
cients{w
n
}are the weights (w
n
> 0). The latter weights
should not be confused with the standard Christoffel numbers{w
n
}from
numerical integrations [5]. We shall also use the completeness relation
∞
W(x)
w
n
W
n
(x)Q
∗
n
(x)Q
m
(x
0
)
δ(x−x
0
) =
W
n
(x) =
.
(2.36)
n=0
This relationship formally retains its form for operatorvalued functions
∞
δ(E1−H) =
W
n
(E)Q
∗
n
(E)Q
m
( H)
(2.37)
n=0
where E is the real energy and H is the Hermitean Hamiltonian operator
( H
†
= H) of the investigated physical system. In (2.37), the Dirac δoperator
represents the socalled density operator for the total energy E and Hamilto
nian
H. There exists the following operator relationship, in the limit ǫ−→0
+
1
(E + iǫ)1−H
1
E1−H
−iπδ(E1−H)
=P
(2.38)
where the symbolPdenotes the usual principle value. Using this expression,
the density operator becomes
δ(E1−H) =−
1
π
Im G(z)
(2.39)
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