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Computing accurate estimates of the Fourier transform of analog signals fromdiscrete data points is important in many fields of science and engineering.The conventional approach of performing the discrete Fourier transform of thedata implicitly assumes periodicity and bandlimitedness of the signal. In thispaper, we use Gaussian process regression to estimate the Fourier transform (orany other integral transform) without making these assumptions. This ispossible because the posterior expectation of Gaussian process regression mapsa finite set of samples to a function defined on the whole real line, expressedas a linear combination of covariance functions. We estimate the covariancefunction from the data using an appropriately designed gradient ascent methodthat constrains the solution to a linear combination of tractable kernelfunctions. This procedure results in a posterior expectation of the analogsignal whose Fourier transform can be obtained analytically by exploitinglinearity. Our simulations show that the new method leads to sharper and moreprecise estimation of the spectral density both in noise-free andnoise-corrupted signals. We further validate the method in two real-worldapplications: the analysis of the yearly fluctuation in atmospheric CO2 leveland the analysis of the spectral content of brain signals.

Integral Transforms from Finite Data: An Application of Gaussian Process Regression to Fourier Analysis