This is a follow up question to Prove $\frac{1}{2} + \cos(x) + \cos(2x) + \dots+ \cos(nx) = \frac{\sin(n+\frac{1}{2})x}{2\sin(\frac{1}{2}x)}$ for $x \neq 0, \pm 2\pi, \pm 4\pi,\dots$
I am looking for the weighted sum of the $\cos(nx)$ series which is on the left side of the equation. The weights are the fourier series of an arbitrary function. More clearly, I am looking for the minimum value of the series. The maximum value is the sum of the weights (fourier coefficients). Can we say anything about the minimum values?
