Conclusion and outlooks - Signal Processing in MRS with Biomedical Applications - page 391

of rational functions could still be too general, because cuts and branch point
singularities might not be so often encountered outside of physics. Even when
cuts and branch points are inherently present in a given problem (e.g., in
particle scattering and in any other systems with interactive dynamics lead
ing to continuous spectra), they could be closely approximated by a sequence
of poles. This leads to a specialized subclass of rational functions called
meromorphic functions whose only singularities are exclusively due to poles
of the denominator function g(u). This presumes analyticity of the numerator
function f(u) which can have zeros, but is otherwise free of singularities.
The subclass of functions that optimally meets these natural demands is
comprised of two different polynomials for the numerator f(u) and denom
inator g(u). With this choice of pure polynomials for both f(u) and g(u),
function R(u) = f(u)/g(u) becomes a quotient of two polynomials, i.e., a
rational polynomial called the Pade approximant, PA. Such a ratio is unique
whenever the function to be modeled is represented by a known Taylor or
Maclaurin series in powers of the given independent variable. This eliminates
ambiguities from the outset in the definition of the PA. Pade approximants
can also be viewed as a generalization of Taylor or Maclaurin polynomials to
the field of rational functions. These rational Pade polynomials are of great
practical usefulness. Among their most important features is acceleration of
slowly converging series or sequences and transformation of divergence into
convergence by the powerful Cauchy concept of analytical continuation.
This versatile Pade strategy represents an excellent approximation method
ology and, moreover, rational functions are no more di cult to employ than
ordinary polynomials like those of Taylor, Maclaurin or Laurent. However,
Pade rational polynomials are infinitely more powerful than any single poly
nomial approximation via Taylor, Maclaurin or Laurent truncated series. Be
ing polynomials, f(u) and g(u) are both analytic and so is their quotient
R(u) = f(u)/g(u), with the exception of poles which are zeros of the de
nominator. For instance, in magnetic resonance spectroscopy, MRS, which
is the main subject of the present book, zeros{u k }of g(u) are complex

valued numbers related to fundamental energies{E k

},
such that the Pade approximant R(u) = f(u)/g(u) can be used to provide the
most adequate model of the system response function for the given external
perturbation. In such a case, R(u) = f(u)/g(u) constitutes a complexvalued
spectrum whose real value is the physical, absorption spectrum. In practice,
we sweep across the real energies or frequencies at which these spectra can
safely be computed. This makes the Pade approximant R(u) = f(u)/g(u) a
welldefined function throughout, even at the zeros{u k }of g(u), since for any
realvalued frequency, we have u = u k , i.e., the values g(u k ) = 0 never occur,
thus making the potential singularity R(u k ) =∞unattainable.
As stated, the importance of rational functions is not limited to mathemat
ics alone. Rather, they have unparalleled applications in all other branches
of science and technology, including industry. This can be easily understood
by reference to the main characteristics of versatile systems considered in iso

}, or frequencies{ω k