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 $$f(t)=\sum _{i=1}^n{b}_i{x}^{2i},b_i\in \mathbb{R}\mathrm{.}$$ 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 $⟨x,y⟩$ with $\cos ({\theta }_{x,y})$, we can reformulate the original problem as: $$\underset{\mu \in P(T)}{\min }{\int }_T{\int }_{T}f(⟨x,y⟩)d\mu (x)d\mu (y)$$ as an equivalent form: $$\underset{\nu \in P([-\pi ,\pi ])}{\min }\sum _{i=1}^n{c}_i{\int }_{-\pi }^{\pi }{\int }_{-\pi }^{\pi }\cos (n(x-y))d\nu (x)d\nu (y)\text{.}$$