MATHEMATICS OF COMPUTATION Volume 73, Number 245, Pages 243–250 S 0025-5718(03)01540-0 Article electronically published on July 14, 2003 A NOTE ON A PAPER BY G. MASTROIANNI AND G. MONEGATO G. CRISCUOLO Abstract. Recently, Mastroianni and Monegato derived error estimates for a numerical approach to evaluate the integral  b a  1 −1 f (x, y) x - y dxdy, where (a, b) ≡ (-1, 1) or (a, b) ≡ (a, -1) or (a, b) ≡ (1,b) and f (x, y) is a smooth function (see G. Mastroianni and G. Monegato, Error estimates in the numerical evaluation of some BEM singular integrals, Math. Comp. 70 2001, 251–267). The error bounds for the quadrature rule approximating the inner integral given in Theorems 3, 4 of that paper are not correct according to the proof. However, following a different approach, we are able to improve the pointwise error estimates given in that paper. 1. Introduction Following a recent numerical approach, Mastroianni and Monegato have sug- gested approximating the integral (1.1) H (f ; y) :=  1 −1 f (x, y) x − y dx, whenever y ∈ (−1, 1) or y/ ∈ (−1, 1) by a quadratue rule of interpolatory type based on the zeros of suitable orthogonal polynomials (see [6]). When y ∈ (−1, 1), the integral H (f ; y) is defined in the Cauchy principal value sense. An accurate calcu- lation of (1.1) may be useful for many applications, for instance, to approximate the two-dimensional integrals of type (1.2)  b a  1 −1 f (x, y) x − y dxdy, where (a, b) ≡ (−1, 1) or (a, b) ≡ (a, −1) or (a, b) ≡ (1,b). Such integrals arise in some applications of Galerkin boundary element methods (see also [6] and the references given therein). Furthermore, the estimate of the error in the numerical approximation of (1.1) can be used in the numerical solution of singular integral equations by a collocation method. Assuming that a symmetric Jacobi weight Received by the editor March 22, 2002. 2000 Mathematics Subject Classification. Primary 41A55; Secondary 65D32, 65N38. Key words and phrases. Singular integrals, error estimates, Lagrange operator, functions of the second kind. c 2003 American Mathematical Society 243