On the convergence of a rule by Monegato for the numerical evaluation of Cauchy principal value integrals (Q1096112)

The authors examine the convergence of an interpolatory type quadrature rule proposed by \textit{G. Monegato} [ibid. 29, 337-354 (1982; Zbl 0485.65017)] for the evaluation of the Cauchy principal value integral \(\int ^{1}_{-1}f(x)(x-t)^{-1}dx.\) This quadrature rule is an interpolatory scheme on \(4m+1\) knots depending on the zeros of Legendre polynomials, with degree of exactness at least 4m. In this work,the uniform convergence in the whole open set (-1,1) for functions f such that \(\int ^{1}_{0}\omega (f;u)u^{-1}du<\infty\) is proved, where \(\omega\) (f;\(\cdot)\) denotes the modulus of continuity. Moreover, some estimates of the remainder, depending on the regularity of the function f, are given.