Asymptotic Properties and Weighted Estimation of Polynomials, Orthogonal on the Nonuniform Grids with Jacobi Weight

Let -1 = eta(0) < eta(1) < eta(2) < ... < eta(N - 1) < eta(N = 1), lambda(N) = max(0 <= j <= 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) <= t(j) <= eta(j+1). In case of integar alpha,beta >= 0 for such discrete orthonormal polynomials (P) over cap (alpha,beta)(n,N)(t) (n - 0 ,..., N-1) asymptotic formula (P) over cap (alpha,beta)(n,N)(t) = (P) over cap (alpha,beta)(n) + upsilon(alpha,beta)(n,N)(t) with n = O(lambda(-1/3)(N)) (lambda(N) -> 0) was obtained, where (P) over cap (alpha,beta)(n)(t) - classical jacobi polynomial, upsilon(alpha,beta)(n,N)(t) - remainder term. As corollary of asymptotic formula it was deduced weighted estimation of (P) over cap (alpha,beta)(n,N)(t) polynomials on segment [-1,1].