  We present a method for deriving a smoothed estimate of the peculiar velocity
field of a set of galaxies with measured circular velocities $\eta\equiv {\rm
log} \Delta v$ and apparent magnitudes $m$. The method is based on minimizing
the scatter of a linear inverse Tully-Fisher relation $\eta= \eta(M)$ where the
absolute magnitude of each galaxy is inferred from its redshift $z$, corrected
by a peculiar velocity field, $M \propto m - 5\log(z-u)$. We describe the
radial peculiar velocity field $u({\bf z})$ in terms of a set of orthogonal
functions which can be derived from any convenient basis set; as an example we
take them to be linear combinations of low order spherical harmonic and
spherical Bessel functions. The model parameters are then found by maximizing
the likelihood function for measuring a set of observed $\eta$. The predicted
peculiar velocities are free of Malmquist bias in the absence of
multi-streaming, provided no selection criteria are imposed on the measurement
of circular velocities. This procedure can be considered as a generalized
smoothing algorithm of the peculiar velocity field, and is particularly useful
for comparison to the smoothed gravity field derived from full-sky galaxy
redshift catalogs such as the IRAS surveys. We demonstrate the technique using
a catalog of ``galaxies" derived from an N-body simulation. Increasing the
resolution of the velocity smoothing beyond a certain level degrades the
correlation of fitted velocities against the velocities calculated from linear
theory methods, which have finite resolution,
