Optimal Learning Rates for Regularized Least-Squares with a Fourier Capacity Condition

We derive minimax adaptive rates for a new, broad class of Tikhonov-regularized learning problems in Hilbert scales under general source conditions. Our analysis does not require the regression function to be contained in the hypothesis class, and most notably does not employ the conventional \textit{a priori} assumptions on kernel eigendecay. Using the theory of interpolation, we demonstrate that the spectrum of the Mercer operator can be inferred in the presence of "tight'' $L^{\infty}$ embeddings of suitable Hilbert scales. Our analysis utilizes a new Fourier capacity condition, characterizes the optimal Lorentz range space of a modified Mercer operator in certain parameter regimes.