Nystrm interpolants based on the zeros of Legendre polynomials for a non-compact integral operator equation

We consider two different Nystrm interpolants for the numericalsolution fo the following singular integral equation arising from a problem of determining the distribution of stressin a thin elastic plate in the vicinity of a cruciform crack.These interpolants originate from the discretization of theintegral by two different quadrature formulas of interpolatorytype based on the zeros of Legendre orthogonal polynomials.The first quadrature is of product type and integrates exactlythe kernel; the second one is the well-known Gauss-Legendreformula. First we derive uniform convergence estimates for the two basicquadrature rules. Then by properly modifying the interpolantassociated with the Gauss-Legendre rule we prove its stabilityand derive for it a uniform error estimate of the type O(n^–4+),>0 as small as we like. We also show that if we had beenable to prove the stability of the first (modified) interpolantwe would have obtained a similar convergence estimate. Finally,for the Gauss-Legendre interpolant we prove that in any closedsubinterval [ 1] (0, 1] the rate of convergence is at leastO(n^–6+). Some numerical results which show the accuracy of our approximantsare also presented.