Nonsymmetric product integration rules for Chebyshev weight functions with Chebyshev abscissae

We study four product integration rules, two for the Chebyshev weight of the first-kind based on the Chebyshev abscissae of the third or fourth-kind, and another two for the Chebyshev weight of the second-kind based again on the Chebyshev abscissae of the third or fourth-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 asymptotically optimal error bounds for these formulae by Peano kernel methods. In addition, the convergence of the quadrature formulae is shown not only for Riemann integrable functions on [ − 1 , 1 ] , but also for functions having a monotonic singularity at one or both endpoints of [ − 1 , 1 ] . Interestingly enough, the rules for the Chebyshev weight of the second-kind based on the Chebyshev abscissae of the third or fourth-kind have the best possible degree of exactness for an interpolatory formula not of Gauss type.