  We show that resonant frequencies of a system of coupled resonators in a
truncated periodic lattice converge to the essential spectrum of corresponding
infinite lattice. We use the capacitance matrix as a model for fully coupled
resonators with long-range interactions in three spatial dimensions. For one-,
two- or three-dimensional lattices embedded in three-dimensional space, we show
that the (discrete) density of states for the finite system converge in
distribution to the (continuous) density of states of the infinite system. We
achieve this by proving a weak convergence of the finite capacitance matrix to
corresponding (translationally invariant) Toeplitz matrix of the infinite
structure. With this characterization at hand, we use the truncated Floquet
transform to introduce a notion of spectral band structure for finite
materials. This principle is also applicable to structures that are not
translationally invariant and have interfaces. We demonstrate this by
considering examples of perturbed systems with defect modes, such as an
analogue of the well-known interface Su-Schrieffer-Heeger (SSH) model.
