Asymptotic properties and weighted estimation of polynomials, orthogonal on the nonuniform grids with Jacobi weight

M. S. Sultanakhmedov

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 14 (2014) no. 1, pp. 38-47

Résumé

Let $-1=\eta_0\eta_1\eta_2\dots\eta_{N-1}\eta_N=1$, $\lambda_N=\max_{0\leq j\leq N-1}(\eta_{j+1}-\eta_j)$. Current work is devoted to investigation of properties of polynomials, orthogonal with Jacobi weight $\kappa^{\alpha,\beta}(t)=(1-t)^\alpha (1+t)^\beta$ on nonuniform grid $\Omega_N=\{t_j\}_{j=0}^{N-1}$, where $\eta_j\leq t_j\leq\eta_{j+1}$. In case of integer $\alpha,\beta\geq0$ for such discrete orthonormal polynomials $\hat P_{n,N}^{\alpha,\beta}(t)$ ($n=0,\ldots,N-1$) asymptotic formula $\hat P_{n,N}^{\alpha,\beta}(t)=\hat P_n^{\alpha,\beta}(t)+\upsilon_{n,N}^{\alpha,\beta}(t)$ with $n=O(\lambda_N^{-1/3})$ ($\lambda_N\to0$) was obtained, where $\hat P_n^{\alpha,\beta}(t)$ – classical Jacobi polynomial, $\upsilon_{n,N}^{\alpha,\beta}(t)$ – remainder term. As corollary of asymptotic formula it was deduced weighted estimation of $\hat P_{n,N}^{\alpha,\beta}(t)$ polynomials on segment $[-1,1]$.