A definiteness criterion for linear functionals and its application to Cauchy principal value quadrature

Let G be a linear functional on Ak[−1, 1] (the class of all functions with an absolutely continuous (k − 1)th derivative), let ξ ∈ R and let G[ξ] be given by G[ξ][ƒ]: = G[(· − ξ)ƒ]. In this situation, a theo Braß and Schmeißer (1981) states that G is definite if G[ξ] is definite. In this paper, we give conditions on G which ensure the definiteness of G[ξ]. An application of our result yields error bounds for a quadrature rule for Cauchy principal value integrals.