Product integration rules for Chebyshev weight functions with Chebyshev abscissae

We study two product integration rules, one for the Chebyshev weight of the first-kind based on the Chebyshev abscissae of the second-kind, and another one constructed the other way around, i.e., relative to the Chebyshev weight of the second-kind and based on the Chebyshev abscissae of the first-kind. The new rules are shown to have positive weights given by explicit formulae. Furthermore, we determine the precise degree of exactness and we compute the variance of the quadrature formulae, we examine their definiteness or nondefiniteness, and we obtain error bounds for these formulae either asymptotically optimal by Peano kernel methods or for analytic functions by Hilbert space techniques. In addition, the convergence of the quadrature formulae is shown not only for Riemann integrable functions on [−1,1], but also, by generalizing a result of Rabinowitz, for functions having a monotonic singularity at one or both endpoints of [−1,1].