  We describe a spectrally-filtered discrete-in-time downscaling data
assimilation algorithm and prove, in the context of the two-dimensional
Navier--Stokes equations, that this algorithm works for a general class of
interpolants, such as those based on local spatial averages as well as point
measurements of the velocity. Our algorithm is based on the classical technique
of inserting new observational data directly into the dynamical model as it is
being evolved over time, rather than nudging, and extends previous results in
which the observations were defined directly in terms of an orthogonal
projection onto the large-scale (lower) Fourier modes. In particular, our
analysis does not require the interpolant to be represented by an orthogonal
projection, but requires only the interpolant to satisfy a natural
approximation of the identity.
