  The extremal values of multivariate trigonometric polynomials are of interest
in fields ranging from control theory to filter design, but finding the
extremal values of such a polynomial is generally NP-Hard. In this paper, we
develop simple and efficiently computable estimates of the extremal values of a
multivariate trigonometric polynomial directly from its samples. We provide an
upper bound on the modulus of a complex trigonometric polynomial, and develop
upper and lower bounds for real trigonometric polynomials. For a univarite
polynomial, these bounds are tighter than existing bounds, and the extension to
multivariate polynomials is new. As an application, the lower bound provides a
sufficient condition to certify global positivity of a real trigonometric
polynomial. We use this condition to motivate a new algorithm for
multi-dimensional, multirate, perfect reconstruction filter bank design. We
demonstrate our algorithm by designing a 2D perfect reconstruction filter bank.
