Mesh-Degree Rigidity for Positive Chebyshev-Fourier Approximants

What the math gives to ML

The paper gives a constructive representation of functions as positive cosine mixtures on a frequency lattice, together with a scaling law linking lattice spacing, polynomial degree, and spectral bandwidth. The transferable asset is the identity Q(cos(dz)) = sum_j a_j cos(jdz) with nonnegative coefficients, which makes the function the Fourier transform of a positive discrete measure with bounded support. In neural networks, this suggests replacing unconstrained Fourier features or oscillatory positional biases with positive lattice spectral mixtures. The paper's asymptotic window provides a principled initialization and regularization regime, although its approximation guarantees must be validated empirically in neural models.