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.