  We present a new method for recovering the cosmological density, velocity,
and potential fields from all-sky redshift catalogues. The method is based on
an expansion of the fields in orthogonal radial (Bessel) and angular (spherical
harmonic) functions. In this coordinate system, peculiar velocities introduce a
coupling of the radial harmonics describing the density field in redshift space
but leave the angular modes unaffected. In the harmonic transform space, this
radial coupling is described by a distortion matrix which can be computed
analytically within the context of linear theory; the redshift space harmonics
can then be converted to their real space values by inversion of this matrix.
Statistical noise is mitigated by regularizing the matrix inversion with a
Wiener filter. The method yields a minimum variance estimate of the density
field in real space. In this coordinate system, the minimum variance harmonics
of the peculiar velocity and potential fields are related to those of the
density field by simple linear transformations. Tests of the method with
simulations of a CDM universe and comparison with previously proposed methods
demonstrate it to be a very promising new reconstruction method for the local
density and velocity field. A first application to the 1.2 Jy IRAS redshift
survey is presented.
