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