Interpolatory quadrature formulae with Chebyshev abscissae of the third or fourth kind

We consider interpolatory quadrature formulae, relative to the Legendre weight function on [-1,1], having as nodes the zeros of the nth-degree Chebyshev polynomial of the third or fourth kind. Szego has shown that the weights of these formulae are all positive. We derive explicit formulae for the weights, and subsequently use them to establish the convergence of the quadrature formulae for functions having a monotonic singularity at one or both endpoints of [-1,1]. Moreover, we generate two new quadrature formulae, by adding 1,-1 to the sets of nodes considered previously, and show that these new formulae have almost all weights positive, exceptions occurring only among the weights corresponding to 1,-1. Also, we determine the precise degree of exactness of all the quadrature formulae in consideration, we obtain asymptotically optimal error bounds for these formulae, and show that almost all of them are nondefinite, exceptions occurring only among the formulae with a small number of nodes

Stieltjes polynomials and related quadrature formulae for a class of weight functions

Consider a (nonnegative) measure d sigma with support in the interval [a, b] such that the respective orthogonal polynomials, above a specific index l, satisfy a three-term recurrence relation with constant coefficients. We show that the corresponding Stieltjes polynomials, above the index 2l - 1, have a very simple and useful representation in terms of the orthogonal polynomials. As a result of this, the Gauss-Kronrod quadrature formulae for da have all the desirable properties, namely, the interlacing of nodes, their inclusion in the closed interval [a,b] (under an additional assumption on d sigma), and the positivity of all weights. Furthermore, the interpolatory quadrature formulae based on the zeros of the Stieltjes polynomials have positive weights, and both of these quadrature formulae have elevated degrees of exactness