  We establish the completeness of some characteristic sets of non-normalizable
modes by constructing fully localized square steps out of them, with each such
construction expressly displaying the Gibbs phenomenon associated with trying
to use a complete basis of modes to fit functions with discontinuous edges. As
well as being of interest in and of itself, our study is also of interest to
the recently introduced large extra dimension brane-localized gravity program
of Randall and Sundrum, since the particular non-normalizable mode bases that
we consider (specifically the irregular Bessel functions and the associated
Legendre functions of the second kind) are associated with the tensor
gravitational fluctuations which occur in those specific brane worlds in which
the embedding of a maximally four-symmetric brane in a five-dimensional anti-de
Sitter bulk leads to a warp factor which is divergent. Since the brane-world
massless four-dimensional graviton has a divergent wave function in these
particular cases, its resulting lack of normalizability is thus not seen to be
any impediment to its belonging to a complete basis of modes, and consequently
its lack of normalizability should not be seen as a criterion for not including
it in the spectrum of observable modes. Moreover, because the divergent modes
we consider form complete bases, we can even construct propagators out of them
in which these modes appear as poles with residues which are expressly finite.
Thus even though normalizable modes appear in propagators with residues which
are given as their finite normalization constants, non-normalizable modes can
just as equally appear in propagators with finite residues too -- it is just
that such residues will not be associated with bilinear integrals of the modes.

High Energy Physics - Theory