The role of quantum mechanics in signal processing - Signal Processing in MRS with Biomedical Applications - page 49

where y T is the transpose matrix of y. Next, a straightforward calculation
with the help of (2.88) yields

n−1 n−1

0 = y T H n ( 0 )y = y r y s r+s

r=0 s=0

n−1 n−1 n−1 n−1 n−1 2

y s x s x r y s x s y r x r y s x s = y r = =

r=0 s=0 s=0 r=0 s=0

n−1

2

y s x s

∴ = 0. (2.89)

s=0

n−1
s=0 y s x s is not permitted, the ob
tained result (2.89) contradicts the assumption det H n ( 0 ) = 0. Hence, the
opposite is true, which proves (2.86) (QED).
When we first replace n by n + 1 in (2.67) and subsequently multiply the
ensuing formula by β n+1 , the following expression is obtained with the use of
the recursion (2.49)

Since this zero norm of the polynomial

β n+1 π n+1 (z) = (z−α n )π n (z)−β n π n−1 (z). (2.90)

This indicates that the denominator{π n (z)}of the errorE

n (z) from (2.66)
can be constructed using the recursion (2.49). Therefore, as the Lanczos
algorithm proceeds, the error at each step can be computed in concert with
the same kind of recursion, as is clear from (2.49) and (2.90). Such an e cient
error analysis is of great practical value in versatile applications of the Lanczos
algorithm.
Because the set{Q r (x)}is a basis, it is possible to expand, e.g., the Green
operator

G(z) from (2.40) in terms of Q n (x) as

∞

γ n (z)≡ η n (z)
w n

G(z) =

γ n (z)Q n ( H)

(2.91)

n=0

where{η n (z)}are the expansion coe cients. The explicit expression for the
coe cients η n (z) can be derived by the standard projection technique. This
procedure shows that the general coe cient η n (z) is given by the Hilbert
transform of Q ∗ n (x)

b

dσ(x)
z−x Q ∗ n (x).

η n (z) = (2.92)

a

Moreover, by comparing (2.67) with (2.92) and using (2.32), we have

η n (z) = π n (z). (2.93)

It should be noted that the coe cients{η n (z)}in (2.91) are independent of
the Hamiltonian

H, which is here present merely via Q n ( H). Thus, as soon as