  We study the relationship between the local density of states (LDOS) and the
conductance variation $\Delta G$ in scanning-gate-microscopy experiments on
mesoscopic structures as a charged tip scans above the sample surface. We
present an analytical model showing that in the linear-response regime the
conductance shift $\Delta G$ is proportional to the Hilbert transform of the
LDOS and hence a generalized Kramers-Kronig relation holds between LDOS and
$\Delta G$. We analyze the physical conditions for the validity of this
relationship both for one-dimensional and two-dimensional systems when several
channels contribute to the transport. We focus on realistic Aharonov-Bohm rings
including a random distribution of impurities and analyze the LDOS-$\Delta G$
correspondence by means of exact numerical simulations, when localized states
or semi-classical orbits characterize the wavefunction of the system.
