arXiv:2611.13930v1 Announce Type: cross Abstract: The uncertainty estimates Gaussian processes output rely on the chosen covariance kernel and its hyperparameters. We study the fixed-domain asymptotics of the maximum likelihood and cross-validation estimators of the lengthscale parameter of the popular infinitely smooth Gaussian kernel when the observations are produced by a fixed univariate response function. No theoretical results for Gaussian lengthscale estimation exist in this setting. We prove lower bounds on the lengthscale estimators that depend on the regularity of the response function, as measured by the rate of decay of its Fourier transform. If the response function lives in the Hilbert space of the Gaussian kernel with lengthscale $\ell$, the lower bounds have a straightforward dependency on $\ell$. For bandlimited functions and functions whose Fourier transforms decay as $\exp(-a \lvert \omega \rvert^\gamma)$ for $a > 0$ and $\gamma > 2$, we prove that the estimators blow up essentially at least as fast as $(\log n)^{(\gamma-2)/(2\gamma)}$ in the number of data points, $n$. Bandlimited functions are identified with $\gamma = \infty$. Unlike analogous results for finitely smooth Mat\'ern kernels, our results make no assumptions on the geometry of the observation points. Numerical experiments illustrate the results.