Wavefield extrapolation by recursive (depth-by-depth) application of a convolutional operator in the frequency-space domain, commonly used for depth migration in a laterally-varying earth, has interesting accuracy and stability properties. We analyze these properties by investigating the operator and its spatial Fourier transform. In particular, we show that the instability caused by spatially truncating the operator can be remedied unconditionally by applying an appropriately chosen spatial taper. However, unconditional stability is gained only at the expense of accuracy. We also identify frequencies and depth extrapolation step sizes for which the problems of accuracy or stability are the most pronounced.