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I guess the following construct will help. The idea is that the set of coefficients which renders a trigonometric polynomial positive (just like the one you have) is a convex set, and is described by a set of LMI constraints. Such a characterization (essentially arising out of Kalman-Yakubovich-Popov Lemma) is routinely used in Control/Sys-Id. Refer to Lemma 2.1 of http://www.ent.mrt.ac.lk/iml/paperbase/TAC%20Collection/TAC/2005/october/7.pdf for the exact details pertaining to your problem. So, basically you end up with: $$ \max ~~A_k~~ \mbox{subject to} ~~ \mbox{convex LMI constraints}, $$ which is a linear SDP and can be solved easily using CVX (or some such solver).