Polynomial potential minimization on the unit circle

In the following, we study the minimization of polynomial potentials $f(t)$ on the unit circle, where the potentials take the form $$f(t) = \sum_{i=1}^n b_i x^{2i}, \quad b_i \in \mathbb{R}.$$ This form arises in the context of truncations of expansions of $p$-frame potentials. One approach to minimize these potentials involves rewriting the integral as a sum of integrals obtained by expanding the potential $f(t) = \sum_{i=1}^n c_i T_i(t)$ in terms of Chebyshev polynomials. By replacing the inner product $\langle x, y \rangle$ with $\cos(\theta_{x, y})$, we can reformulate the original problem as: $$\min_{\mu \in P(T)} \int_T \int_T f(\langle x, y \rangle) d\mu(x) d\mu(y)$$ as an equivalent form: $$\min_{\nu \in P([-\pi, \pi])} \sum_{i=1}^n c_i \int_{-\pi}^\pi \int_{-\pi}^\pi \cos(n(x - y)) d\nu(x) d\nu(y)$$.