Polynomials, orthogonal on grids from unit circle and number axis

In current paper we investigate the asymptotic properties of polynomials, orthogonal on arbitrary (not necessarily uniform) grids from an unit circle or segment $[-1,1]$. When the grid of nodes $\Omega_N^T=\left\{e^{i\theta_0},e^{i\theta_1},\ldots,e^{i\theta_{N-1}}\right\}$ belongs to the unit circle $|w|=1$ we consider polynomials $\varphi_{0,N}(w),\varphi_{1,N}(w),\ldots,$ $\varphi_{N-1,N}(w)$, orthogonal in the following sense:
$$\frac1{2\pi}\int\limits_{-\pi}^\pi \varphi_{n,N}(e^{i\theta})\overline{\varphi_{m,N}(e^{i\theta})}\,d\sigma_N(\theta)=$$

$$\frac1{2\pi}\sum\limits^{N-1}_{j=0} \varphi_{n,N}(e^{i\theta_j})\overline{\varphi_{m,N}(e^{i\theta_j})} \Delta\sigma_N(\theta_j)=\delta_{nm},$$
where $\Delta\sigma_N(\theta_j)=\sigma_N(\theta_{j+1})-\sigma_N(\theta_j), j=0,\ldots,N-1$. In case, when $\Delta\sigma_N(\theta_j)=h(\theta_j)\Delta\theta_j$, the asymptotic formula for $\varphi_{n,N}(w)$ is established, which in turn, used for investigation of asymptotic properties of polynomials which are orthogonal on grids from $[-1,1]$.