An asymptotic formalism developed earlier for the case of observations away from the epicenter and its anitpode is here extended to near-source and near-antipodal observations. This formalism is valid in the limit of large angular orders and for smooth models of the Earth. We show that for a given dispersion branch the variation of the eigenfrequency shift as a function of angular order l is smooth to order 1/l, contrary to the case of nonpolar observations. If the source is isotropic, this frequency shift depends on the even poloidal part of lateral heterogeneity, as seen with respect to the source-receiver axis. For a more general moment tensor source, terms s even, m=¿2 are also present, weighted by the radiation pattern of the source. In the case of propagating wave trains we show that the phase shift at the antipode is &pgr;/2 for individual wave packets (R1R2, R3R4, ...) and that the remaining phase term can be converted into a phase velocity related to the global Earth structure. The measured phase velocities should be equal from one wave packet to the next and are directly convertible into the corresponding normal mode frequency shifts. We illustrate this using observations from the GEOSCOPE network. We thus show that antipodal observations of free oscillations eigenfrequencies or alternatively of individual wave trains can be used to put additional constraints on the even-order large-scale lateral heterogeneity in the Earth's mantle. ¿ American Geophysical Union 1988