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

摘要

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