Bounding multivariate trigonometric polynomials

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 univariate 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.