  We propose spectral methods for long-term forecasting of temporal signals
stemming from linear and nonlinear quasi-periodic dynamical systems. For linear
signals, we introduce an algorithm with similarities to the Fourier transform
but which does not rely on periodicity assumptions, allowing for forecasting
given potentially arbitrary sampling intervals. We then extend this algorithm
to handle nonlinearities by leveraging Koopman theory. The resulting algorithm
performs a spectral decomposition in a nonlinear, data-dependent basis. The
optimization objective for both algorithms is highly non-convex. However,
expressing the objective in the frequency domain allows us to compute global
optima of the error surface in a scalable and efficient manner, partially by
exploiting the computational properties of the Fast Fourier Transform. Because
of their close relation to Bayesian Spectral Analysis, uncertainty
quantification metrics are a natural byproduct of the spectral forecasting
methods. We extensively benchmark these algorithms against other leading
forecasting methods on a range of synthetic experiments as well as in the
context of real-world power systems and fluid flows.
