Asymptotic properties of zeros of orthogonal trigonometric polynomials of half-integer orders

For systems of orthogonal trigonometric polynomials of half-integer orders obtained by the Schmidt orthogonalization of the sequences $\cos(1/2)\tau$, $\sin(1/2)\tau$, $\cos(3/2)\tau$, $\sin(3/2)\tau$, $\cos(5/2)\tau$, $\sin(5/2)\tau,\dots$ and $\sin(1/2)\tau$, $\cos(1/2)\tau$, $\sin(3/2)\tau$, $\cos(3/2)\tau$, $\sin(5/2)\tau$, $\cos(5/2)\tau,\dots$ in the measure $d\sigma(\tau)$ on $[0,2\pi]$, we study the connections with the system of polynomials that is orthogonal on the unit circle in the same measure. An asymptotic formula is obtained for zeros of a trigonometric polynomial of half-integer order that is orthogonal with an even weight satisfying the Bernstein–Szego condition.