The role of quantum mechanics in signal processing - Signal Processing in MRS with Biomedical Applications - page 39

simultaneously present in the projector π k . Here, we use the name Schrodinger
operator for ω1− ˆ because this operator originates from the Schrodinger
equation ˆ |Υ) = ω|Υ) which can alternatively be written as the secular or
characteristic equation (ω1− ˆ )|Υ) = 0. The inverse operator (ω1− ˆ ) −1 is
in a special category called resolvent operators. Naturally, (2.30) can exist
on its own as the definition of the Green operator from the outset without
any derivation. Therefore, if (2.30) is considered as the definition of G(ω), it
is useful to replace the unity operator by the closure (2.14). The subsequent
use of the Green eigenvalue problem (2.29) yields the spectral decomposition
(2.21) without recourse to the Fourier integral (2.26).
Importantly, when using the Green function from (2.30), which is obtained
by the above derivation, then G(ω) is recognized as the operator Pade ap
proximant (OPA) [121] (there is also the matrix Pade approximant (MPA)
[122]). Generally, the left

K and the right B − K A L variants of the OPA
yield the same OPA through A L B − K = B − K A L . This allows one to drop the
adjective left or right associated with OPA. In fact, the equality between the
left and the right OPA is easily established from their definitions that are of
the same type as the scalar PA, except for the operatorvalued polynomial
coe cients in A L and B K . These notions also apply to (2.30) because of the
presence of the unity operator for A (the other operator B is ω1− ˆ ). The
OPA structure of the exact Green operator, which is a more fundamental
quantity than the associated scalar, implies that in every realization, the cor
responding exact Green function will necessarily be the scalar PA. Hence, it
is not surprising that the main building block of, e.g., the Heisenberg picture
of quantum mechanics, the scattering operator or the S−operator must be in
the mathematical form of a rational function, as shown first by Hu [123].
The above analysis demonstrates that the Pade approximant is actually el
evated to the status of being an inherent, fundamental formalism of quantum
mechanics per se from the outset. As such, the PA represents an insepara
ble part of the conceptual foundation of quantum mechanics, and not only
an exceptionally successful computational method. This remark is made in
the abstract theoretical framework with an underlying total generality and
with no reference to any particular algorithmic methodology. This status
of the Pade approximant justifies the previous concrete derivations using a
number of the leading quantummechanical methods, that the Pade approx
imant (operatorvalued and/or scalar) is equivalent to the Schwinger varia
tional principle, variationaliteration methods and the Fredholm determinant,
to name only a few of the leading theories [5]. In particular, the establish
ment of the variational nature of the Pade approximant is the critical feature
of this method especially related to the stability of spectral estimation as well
as to the feasibility of obtaining the upper and lower bounds to the computed
spectral parameters of generic systems.
Regarding the computational performance, judicious implementation of the
theoretical formalism of the Green function is significant in itself, on top

A L B −1