SOME NODES MATRICES APPEARING IN THE NUMERICAL-ANALYSIS FOR SINGULAR INTEGRAL-EQUATIONS

Let {p(m)(w)} be the sequence of Jacobi polynomials corresponding to the weight w(x) = (1 - x)alpha(1 + x)beta, 0 < alpha, beta < 1. Denote by x(k)(w) = cos theta(m,k)(w), k = 1,...,m, the zeros of p(m)(w). If alpha + beta = 0, then the estimates c1m-1 less-than-or-equal-to\theta(m,k)(w-1)\less-than-or-equal-to c2m-1 hold uniformly with respect to k and m (Theorem 2.1). A similar result holds in the case alpha + beta = 1. Moreover, approximation properties of the Lagrange polynomial interpolating a function at the zeros of p(m)(w)p(m)(w-1) and at +/- 1 are studied (Theorem 2.3). These results have a crucial role in the error analysis of quadrature methods for solving Cauchy singular integral equations on the interval (-1, 1) [6].