  This paper investigates existence of the nonstandard Pade approximants
introduced by Cherkaev and Zhang in J. Comp. Phys. 2009 for approximating the
spectral function of composites from effective properties at different
frequencies. The spectral functions contain microstructure information. Since
this reconstruction problem is ill-posed [9], the well-performed Pade approach
is noteworthy and requires further investigations. In this paper, we validate
the assumption that the effective dielectric component of interest can be
approximated by Pade approximants whose denominator has nonzero power one term.
We refer to this as the nonstandard Pade approximant, in contrast to the
standard approximants with nonzero constant terms. For composites whose
spectral function assumes infinitely many different values, the proof is
carried by using classic results for Stieltjes functions. For those with
spectral functions having only finitely many different values, we prove the
results by utilizing a special product decomposition of the coefficient matrix
of the Pade system. The results in this paper can be considered as an extension
of the Pade theory for Stieltjes functions whose spectral function take
infinitely many different values to those taking only finitely many values. In
the literature, the latter is usually excluded from the definition of Stieltjes
functions because they correspond to rational functions, hence convergence of
their Pade approximants is trivial. However, from an inverse problem point of
view, our main concern is the existence of the nonstandard Pade approximants,
rather than their convergence. The results in this paper provide a mathematical
foundation for applying the Pade approach for reconstructing the spectral
functions of composites whose microstructure is not a priori known.
