On-axis and far-field series expansions are developed for the sound pressure due to an arbitrary, circular symmetric velocity distribution on a flat radiator in an infinite baffle. These expansions are obtained by expanding the velocity distributions in terms of orthogonal polynomials R(/a)=Pn(2(/a)2-1) with Pn the Legendre polynomials. The terms R give rise to a closed-form expression for the pressure on-axis as well as for the far-field pressure. Furthermore, for a large number of velocity profiles, including those associated with the rigid piston, the simply supported radiator, and the clamped radiators as well as Gaussian radiators, there are closed-form expressions for the required expansion coefficients. In particular, for the rigid, simply supported, and clamped radiators, this results in explicit finite-series expressions for both the on-axis and far-field pressures. In the reverse direction, a method of estimating velocity distributions from (measured) on-axis pressures by matching in terms of expansion coefficients is proposed. Together with the forward far-field computation scheme, this yields a method for far-field loudspeaker assessment from on-axis data (generalized Keele scheme). The forward computation scheme is extended to dome-shaped radiators with arbitrary velocity distributions.