  We propose a novel class of Gaussian processes (GPs) whose spectra have
compact support, meaning that their sample trajectories are almost-surely band
limited. As a complement to the growing literature on spectral design of
covariance kernels, the core of our proposal is to model power spectral
densities through a rectangular function, which results in a kernel based on
the sinc function with straightforward extensions to non-centred (around zero
frequency) and frequency-varying cases. In addition to its use in regression,
the relationship between the sinc kernel and the classic theory is illuminated,
in particular, the Shannon-Nyquist theorem is interpreted as posterior
reconstruction under the proposed kernel. Additionally, we show that the sinc
kernel is instrumental in two fundamental signal processing applications:
first, in stereo amplitude modulation, where the non-centred sinc kernel arises
naturally. Second, for band-pass filtering, where the proposed kernel allows
for a Bayesian treatment that is robust to observation noise and missing data.
The developed theory is complemented with illustrative graphic examples and
validated experimentally using real-world data.
