Knowledge of second-order statistics of channels (e.g. in the form of covariance matrices) is crucial for the acquisition of downlink channel state information (CSI) in massive MIMO systems operating in the frequency division duplexing (FDD) mode. Current MIMO systems usually obtain downlink covariance information via feedback of the estimated covariance matrix from the user equipment (UE), but in the massive MIMO regime this approach is infeasible because of the unacceptably high training overhead. This paper considers instead the problem of estimating the downlink channel covariance from uplink measurements. We propose two variants of an algorithm based on projection methods in an infinite-dimensional Hilbert space that exploit channel reciprocity properties in the angular domain. Performance are evaluated via Monte Carlo simulations, and the proposed schemes are shown to outperform current state-of-the art solutions in terms of accuracy and complexity, for typical array geometries and duplex gaps.