The accuracy and efficiency of the adaptive Fourier integration approach depend on several factors related to the properties of the spectral density. The theoretical limits of accuracy are primarily determined by the smoothness and decay properties of the spectral density. For slowly decaying spectral densities or those with singularities, the accuracy may be limited by numerical precision and the ability to resolve oscillations in the integrand. The efficiency of the approach is influenced by the choice of quadrature rule, the panel selection heuristic, and the computational cost of the NUFFT.
The efficiency of the method is also affected by the Nyquist-informed panel selection heuristic, which determines the size of the panels in the spectral domain based on the frequency content of the spectral density. By adaptively choosing panel sizes, the method can focus computational resources where they are most needed, improving efficiency. Additionally, the efficiency of the approach is influenced by the computational cost of the NUFFT, which accelerates the computation of the Fourier integrals.