A periodic pulse train of period N, and having the value P(t) at time t, is generated according to

P(t) = | A₀ + A₁W + A₁W² + ... + A_N-1W^N-1 | ²

where the coefficients A_j are complex numbers, W=exp(i2πt/N) (i=√(-1)), and N is a fixed positive integer. The pulse train is thus the square of the absolute value of a complex "trigonometric polynomial."

The pulse train engineer desires the pulse train to have the highest peak value possible, subject to the constraint that the area S of one period of the pulse train has a prescribed value. The engineer suspects that there is a smallest constant K such that no matter how the A_j are chosen, P(t)<=KS for all t. Prove this for him if you can, and determine if equality is possible for some t when S>0 (which would make the engineer's desire fully realizable).