We first establish spectral convergence of the random feature method (RFM) for multidimensional targets belonging to Sobolev, Gevrey, ultra‑analytic, and band‑limited classes. The analysis provides high‑probability approximation estimates in the interpolation scale generated by a kernel integral operator. A single event determined solely by the sampled features yields a random space that approximates every target inside a prescribed source ball; moreover, for each target a single coefficient vector defines an approximant that attains spectral accuracy simultaneously in all admissible error norms. For both regularity‑adapted frequency distributions and uniform distributions on expanding frequency windows, the resulting rates range from super‑exponential to algebraic, depending on the target’s regularity. We then derive abstract error estimates for strong‑ and weak‑form RFM discretizations, converting the preceding approximation bounds into convergence results for multidimensional second‑order elliptic boundary‑value and eigenvalue problems. Finally, for random feature matrices (RFMtxs) we prove super‑exponential singular‑value decay with Fourier features and exponential decay with $\tanh$ features, together with corresponding lower bounds on the condition number. The analysis reveals a common mechanism: the same spectral approximation that yields high accuracy also drives severe ill‑conditioning.
Review: This study offers a comprehensive theoretical foundation for the outstanding performance of RFM on smooth high‑dimensional problems, while highlighting the inherent ill‑conditioning that must be addressed in practical implementations.