The role of quantum mechanics in signal processing - Signal Processing in MRS with Biomedical Applications - page 53

•It can be used for systems with discrete energies as well as for resonances
and scattering,

•It provides the integrated density of states, i.e., the IDOS, eigenvalues
E k and the associated eigenvectors|Υ k of the Schrodinger eigenvalue
problem without ever solving this problem explicitly,

•The computational and storage costs are attractive, since they scale
linearly with the size of the considered system,

•It is suitable for parallel processing.

2.2.2 Classical polynomials
For the reason of having a more general presentation, the expansion method
from subsection 2.2.1 is concerned with the space of nonclassical, Lanczos
polynomials{P n (x),Q n (x)}. Nevertheless, the whole analysis and the corre
sponding conclusions also hold true for classical polynomials. This is the case
because all the classical polynomials are orthogonal polynomials and, there
fore, they satisfy the same threeterm recursion (2.49) with the unchanged
definition of the coupling constants α n and β n from (2.50) and (2.51). The
sole difference is that α n and β n can be obtained in their analytical forms for
all the classical polynomials. Moreover, the Christoffel numbers{w n }for clas
sical polynomials are computed from the same formula (2.72). Furthermore,
the expression (2.53) can be checked analytically to be valid.
Remarkably, for classical polynomials, the error or remainderE n (z) from
(2.65) can also be calculated analytically. For instance, if Q n (x) is the Cheby
shev polynomial T n (x) of the first kind, then the polynomial P n (x) is the
Chebyshev polynomial U n (x) of the second kind

sin([n + 1]ϕ)
sin ϕ

ϕ = cos −1 (z). T n (z) = cos(nϕ) U n (z) = (2.114)

In such a case, we have from (2.66) that

(z) = π (C n (z)
T n (z)

E (C)
n

(2.115)

where π (C) (z) is given by

n

1 + δ n,0
2

U n−1 (z)− i π (C)

n

√

(z) = π

1−z 2 T n (z)

(2.116)

with δ n,0 being the Kronecker δsymbol. It is also possible to derive a sim
ple analytical expression forE

n (z) if Q n (x) is the Hermite H n (x), Jacobi

P (α,β)

(x), Legendre and Laguerre polynomial. For instance, we can write

n

(E) = π (H n (E)
H n (E)

π (J n (E)
P (α,β)

E (H)
n

E (J)
n

(E) = (2.117)

(E)

n