Data file. The two-column file cue_data.csv [25 KB]
tabulates the power spectrum \(S_\infty (\omega) \)
specified in Theorem 1.2
of the paper
RK-2022 . The first column contains \( 999 \) rescaled frequencies
\( \tilde\omega = \omega/2\pi \) in the interval from
\( 5\times 10^{-4} \) to \(0.5 - 5\times 10^{-4} \) with the increment \( 5\times 10^{-4}\). The second column
contains the power spectrum computed numerically to an absolute and relative errors better than
\( \delta_{\rm abs} = 10^{-6}\) and \( \delta_{\rm rel} = 10^{−5}\), respectively.
Technical remarks. For frequencies away from the edges \( \tilde\omega = 0 \) and
\( \tilde\omega_{\rm Ny} = 1/2 \),
the power spectrum was computed with an ODE solver for the fifth Painlevé transcendent. At very small frequencies,
the data were produced from the small-\( \omega \) expansion proved in Proposition 4.10. Close to the Nyquest frequency
\( \tilde\omega_{\rm Ny} = 1/2 \),
a numerical solution to the fifth Painlevé transcendent becomes unstable due to the poles of \( \sigma_0 ( \lambda;\zeta ) \)
in the complex \( \lambda \)-plane close to the real axis.
There, an evaluation of the power spectrum was based on the Bornemann code
[F. Bornemann, «On the numerical evaluation of Fredholm determinants». Math. Comp. 79, 871 (2010)].