  Computing accurate estimates of the Fourier transform of analog signals from
discrete data points is important in many fields of science and engineering.
The conventional approach of performing the discrete Fourier transform of the
data implicitly assumes periodicity and bandlimitedness of the signal. In this
paper, we use Gaussian process regression to estimate the Fourier transform (or
any other integral transform) without making these assumptions. This is
possible because the posterior expectation of Gaussian process regression maps
a finite set of samples to a function defined on the whole real line, expressed
as a linear combination of covariance functions. We estimate the covariance
function from the data using an appropriately designed gradient ascent method
that constrains the solution to a linear combination of tractable kernel
functions. This procedure results in a posterior expectation of the analog
signal whose Fourier transform can be obtained analytically by exploiting
linearity. Our simulations show that the new method leads to sharper and more
precise estimation of the spectral density both in noise-free and
noise-corrupted signals. We further validate the method in two real-world
applications: the analysis of the yearly fluctuation in atmospheric CO2 level
and the analysis of the spectral content of brain signals.
