Motivated by classical results of approximation theory, we define an Hermite-type interpolation in terms of $n$-dimensional subspaces of the space of $n$ times continuously differentiable functions. In the main result of this paper, we establish an error term in integral form for this interpolation in the case when the $n$-dimensional subspace is the kernel of an $n$th order linear differential operator with constant coefficients. Several corollaries are deduced illustrating the applicability of this result.