  A computationally inexpensive k.p-based interpolation scheme is developed
that can extend the eigenvalues and momentum matrix elements of a sparsely
sampled k-point grid into a densely sampled one. Dense sampling, often required
to accurately describe transport and optical properties of bulk materials, can
be demanding to compute, for instance, in combination with hybrid functionals
in density functional theory (DFT) or with perturbative expansions beyond DFT
such as the $GW$ method. The scheme is based on solving the k.p method and
extrapolating from multiple reference k-points. It includes a correction term
that reduces the number of empty bands needed and ameliorates band
discontinuities. We show how the scheme can be used to generate accurate band
structures, density of states, and dielectric functions. Several examples are
given, using traditional and hybrid functionals, with Si, TiNiSn, and Cu as
model materials. We illustrate that d-electron and semi-core states, which are
particular challenging for the k.p method, can be handled with the correction
scheme if the sparse grid is not too sparse.
