  We study the density-density correlation function $G({\bf r},{\bf r}')$ in
the interfacial region of a fluid (or Ising-like magnet) with short-ranged
interactions using square gradient density functional theory. Adopting a simple
double parabola approximation for the bulk free-energy density, we first show
that the parallel Fourier transform $G(z,z';q)$ and local structure factor
$S(z;q)$ separate into bulk and excess contributions. We attempt to account for
both contributions by deriving an interfacial Hamiltonian, characterised by a
wavevector dependent surface tension $\sigma(q)$, and then reconstructing
density correlations from correlations in the interface position. We show that
the standard crossing criterion identification of the interface, as a surface
of fixed density (or magnetization), does not explain the separation of
$G(z,z';q)$ and the form of the excess contribution. We propose an alternative
definition of the interface position based on the properties of correlations
between points that "float" with the surface and show that this describes the
full $q$ and $z$ dependence of the excess contributions to both $G$ and $S$.
However, neither the "crossing-criterion" nor the new "floating interface"
definition of $\sigma(q)$ are quantities directly measurable from the total
structure factor $S^{tot}(q)$ which contains additional $q$ dependence arising
from the non-local relation between fluctuations in the interfacial position
and local density. Since it is the total structure factor that is measured
experimentally or in simulations, our results have repercussions for earlier
attempts to extract and interpret $\sigma(q)$.
