The role of quantum mechanics in signal processing - Signal Processing in MRS with Biomedical Applications - page 46

While arriving to the expression (2.66), a division by the constant quotient
β 1 / 0 was made. This is allowed, since β 1 = 0 and 0 = 0. Employing (2.32)

and (2.67), we have

π ∗ n (z) = π n (z ∗ ). (2.68)

This is reminiscent of the relation (2.41). Moreover, (2.68) can also be ob
tained from the complex conjugate counterpart of (2.67) by using (2.32) and
(2.41). The derived expression (2.66) for the errorE n (z) is remarkable because
the function Q n (z)E

n (z) coincides precisely with the Hilbert transform of the
polynomial Q n (x) in the vector space defined by the measure dσ(x) and the
asymmetric scalar product for x∈[a,b].
The polynomial quotientR

n (x) from (2.64) is a meromorphic function.

Therefore,R

n (x) can be given by its spectral representation, which is a linear
combination of the Heaviside partial fractions

n

w k
x−x k

R

n (x) =

. (2.69)

k=1

Here,{x k }are the zeros of the polynomials Q n (x), whereas{w k }are the

corresponding Christoffel numbers. Explicitly,{w k }can be obtained by means

of the Cauchy residue theorem applied toR

n (x)

w k = lim
x→x k

(x−x k )R n (x) (2.70)

w k = 0
β 1

P n (x k )
Q ′ n (x k )

= 0
β n

1
Q n−1 (x k )Q ′ n (x k )

(2.71)

β 1
β n Q n−1 (x k )

P n (x k ) = Q n (x k ) = 0 (2.72)

where Q ′ n (x) = (d/dx)Q n (x). The formula (2.71) can also be derived through
the Christoffel-Darboux formula [5]

n−1

Q m (x)Q m (y) = β n Q n (x)Q n−1 (y)−Q n−1 (x)Q n (y)
x−y

. (2.73)

m=0

Summation of all the terms{w k /(x−x k )}on the rhs of (2.69) leads to the
quotient of the two polynomials ( 0 /β 1 )P n (x)/Q n (x). This result represents
the Pade approximant (2.64) from which we started. Hence consistency.
The formulae (2.65) and (2.66) show that it is possible to compute the
errorE n (z) with an absolute certainty whenever the Stieltjes integral (2.62)
is represented by the rational polynomialR n (z) from (2.64). This can be

achieved by the exact Gausstype numerical quadrature rule

n

b

dσ(x)
z−x = lim

w k
z−x k .

(2.74)

n→∞

a

k=1