Basic tasks of signal processing in spectroscopy - Signal Processing in MRS with Biomedical Applications - page 10

i.e., the Pade approximant. Precisely the same response function, which is
equivalently called the Green function, is one of the major methods in quan
tum mechanics, as well. It is remarkable that these two seemingly distant
disciplines share the common Pade approximant as one of their main, multi
faceted work strategies.
In fact, many among the existing major algorithms are equivalent or can
be reduced to the Pade approximant, e.g., Frobenieus' normal forms, Krylov
Danielevskii expansion method, tridiagonal Jacobi J−matrix, Shanks' trans
form, Wynn's ε−algorithm, autoregressive moving average, decimated sig
nal diagonalization, continued fractions, Lanczos nearest neighbor approxima
tion (or minimal iterations), Stiltjes' power moments, Lowdin's partitioning
and inner projections, Willer's modified moments, Rutishauser's quotient
difference, Gordon's productdifference, HaydockHeineCullumWilloughby's
recursive residue generation method, variationiteration method, Schwinger's
variational principle, the socalled N/D (numerator/denominator) method,
Fredholm's determinants, method of finiterank separable potentials, etc. [5].
Moreover, the Pade approximant P/Q has one unparalleled advantage over
all these or other unmentioned methods in signal processing. This is the avail
ability of the closed, analytical expressions derived by Belkic [5, 34] for the
general expansion coe cients of the numerator P and denominator Q poly
nomials of any degree, continued fractions, as well as for the parameters of
the quotientdifference, productdifference and Lanczos' algorithms (coupling
constants, eigenfunctions, including their norms, etc.). This is of paramount
importance because Padebased signal processing from explicit formulae can
be used to benchmark the corresponding numerical algorithms for otherwise
mathematically illconditioned inverse problems, such as the quantification
problem in MRS and elsewhere.
On a more general note, in addition to unifying so many apparently dif
ferent computational algorithms into the general Pade formalism of rational
functions, the Pade approximant has an unprecedented status in theoretical
physics for certain special reasons. This is the case because the Pade approx
imant is known to have rescued the working status of several entire theories.
The most illustrative examples that lend support to this assertion are the
BrillouinWigner perturbation theory and the theory of strong interactions
in elementary particle physics. The RayleighRitz perturbation theory has
good convergence properties. It is a wellestablished framework, which for
nondegenerate states predicts accurate discrete energies with estimates for
the related bounds. To degenerate states, however, one needs to apply the
BrillouinWigner perturbation theory, which is plagued by divergences that
cast doubt upon the usefulness and the prospect of the whole theory. Here,
the Pade approximant played a direct role to resum the divergent perturbation
expansions and, moreover, to find bounds to the exact energies of degenerate
systems. In the theory of strong interactions, the only working method is the
perturbation expansion, for which much effort has to be invested for compu
tations of largerorder Feynman diagrams. But when such hardearned per