Approximation theory

Daghestan Electronic Mathematical Reports: Issue 1 (2014)

Polynomials, orthogonal on grids from unit circle and number axis

UDK: 517.538

Pages: 1 - 55

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 $\Delta\sigma_N(\theta_j)=h(\theta_j)\Delta\theta_j$ for $\varphi_{n,N}(w)$ 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 $[-1,1]$.

Keywords: unit circle, number axis, polynomials orthogonal on grids, asymptotics.

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