Exact spectral allocation in Fourier-feature neural tangent kernels

Abstract Fourier features determine the kernel spectrum of coordinate networks, yet bandwidth and harmonic support leave the learning rates of individual frequencies unspecified. We study fixed, finite integer Fourier dictionaries on the torus. For a dot-product kernel with nonnegative summable power-series coefficients, each Fourier eigenvalue is a mixture of signed lattice-walk probabilities. This separates architecture-dependent coefficients from dictionary-dependent multiplicities and yields exact support, finite-rank approximations with trace-norm error bounds, and certified comparisons of spectral design objectives. For the two-layer bias-free rectified linear unit (ReLU) neural tangent kernel (NTK), we characterize an even-walk subgroup that can exclude frequencies inside the generated lattice and derive the leading one-dimensional spectral tail. Two dictionaries matched in cardinality, maximum frequency, and base second moment have identical leading tails but a 19.51-fold difference in their mode-9 eigenvalues; finite-network decay rates track this difference. Across three target frequencies and 108 dictionary--target pairs, spectral predictions have rank correlations of 0.977--0.992 with fitted rates. Under a four-frequency budget for twenty target modes, spectral selection reduces off-grid finite-network mean squared error (MSE) by 12.3% relative to direct top-energy selection on a mixed-spectrum signal. Fifty runs on five natural-image crops show close agreement with finite-grid MSE predictions and larger deviations in individual mode rates. These results quantify how a fixed Fourier dictionary allocates kernel mass and predicts learning in the NTK regime.