Generalization Error of Minimum Weighted Norm and Kernel Interpolation

We study the generalization error of functions that interpolate prescribed data points and are selected by minimizing a weighted norm. Under natural general conditions, we prove both interpolants their errors converge as number parameters grows, limiting interpolant belongs to reproducing kernel Hilbert space. This rigorously establishes an implicit bias minimum norm interpolation explains why minimization may either benefit or suffer from over-parameterization. As special cases this theory, trigonometric polynomials spherical harmonics. Our approach is deterministic approximation theory viewpoint, opposed statistical random matrix one.