Mesh-Degree Rigidity for Positive Chebyshev-Fourier Approximants

Let $d_k>0$ and let $Q_k$ be a polynomial of degree $r_k$ with nonnegative Chebyshev coefficients. We study locally uniform limits of $f_k(z)=C_kQ_k(\cos(d_kz))$, where $C_k>0$. If $f_k\to F$ locally uniformly and $F(0)>0$, the associated positive lattice measures converge weakly, their second momen