In this paper, we consider the problem of active learning in trigonomet(cid:173) ric polynomial networks and give a necessary and sufficient condition of sample points to provide the optimal generalization capability. By ana(cid:173) lyzing the condition from the functional analytic point of view, we clarify the mechanism of achieving the optimal generalization capability. We also show that a set of training examples satisfying the condition does not only provide the optimal generalization but also reduces the compu(cid:173) tational complexity and memory required for the calculation of learning results. Finally, examples of sample points satisfying the condition are given and computer simulations are performed to demonstrate the effec(cid:173) tiveness of the proposed active learning method.