  Exploiting spectral properties of symmetric banded Toeplitz matrices, we
describe simple sufficient conditions for positivity of a trigonometric
polynomial formulated as linear matrix inequalities (LMI) in the coefficients.
As an application of these results, we derive a hierarchy of convex LMI inner
approximations (affine sections of the cone of positive definite matrices of
size $m$) of the nonconvex set of Schur stable polynomials of given degree $n <
m$. It is shown that when $m$ tends to infinity the hierarchy converges to a
lifted LMI approximation (projection of an LMI set defined in a lifted space of
dimension quadratic in $n$) already studied in the technical literature.
