  The problem of recovering a signal from its low frequency components occurs
often in practical applications due to the lowpass behavior of many physical
systems. Here we study in detail conditions under which a signal can be
determined from its low-frequency content. We focus on signals in
shift-invariant spaces generated by multiple generators. For these signals, we
derive necessary conditions on the cutoff frequency of the lowpass filter as
well as necessary and sufficient conditions on the generators such that signal
recovery is possible. When the lowpass content is not sufficient to determine
the signal, we propose appropriate pre-processing that can improve the
reconstruction ability. In particular, we show that modulating the signal with
one or more mixing functions prior to lowpass filtering, can ensure the
recovery of the signal in many cases, and reduces the necessary bandwidth of
the filter.
