We derive novel bounds for the performance of algorithms that estimate the downlink covariance matrix from the uplink covariance matrix in frequency division duplex (FDD) massive multiple-input multiple-output (MIMO) systems. The focus is on algorithms that use estimates of the angular power spectrum as an intermediate step. Unlike previous results, the proposed bounds follow from simple arguments in possibly infinite dimensional Hilbert spaces, and they do not require strong assumptions on the array geometry or on the propagation model. Furthermore, they are suitable for the analysis of set-theoretic methods that can efficiently incorporate side information about the angular power spectrum. This last feature enables us to derive simple techniques to enhance set-theoretic methods without any heuristic arguments. In particular, we show that the performance of a simple algorithm that requires only a simple matrix-vector multiplication cannot be improved significantly in some practical scenarios, especially if coarse information about the support of the angular power spectrum is available.