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In this paper, we consider the Gauss-Kronrod quadrature formulas for a modified Chebyshev weight. Efficient estimates of the error of these Gauss-Kronrod formulae for analytic functions are obtained, using techniques of contour integration that were introduced by Gautschi and Varga (cf. Gautschi and Varga SIAM J. Numer. Anal. 20, 1170-1186 1983). Some illustrative numerical examples which show both the accuracy of the Gauss-Kronrod formulas and the sharpness of our estimations are displayed. Though for the sake of brevity we restrict ourselves to the first kind Chebyshev weight, a similar analysis may be carried out for the other three Chebyshev type weights; part of the corresponding computations are included in a final appendix.

In the recent paper Notaris (Numer. Math., 142:129-147,2019) it has been introduced a new and useful class of nonnegative measures for which the well-known Gauss-Kronrod quadrature formulae coincide with the generalized averaged Gaussian quadrature formulas. In such a case, the given generalized averaged Gaussian quadrature formulas are of the higher degree of precision, and can be numerically constructed by an effective and simple method; see Spalevic (Math. Comp., 76:1483-1492,2007). Moreover, as almost immediate consequence of our results from Spalevic (Math. Comp.,76:1483-1492,2007) and that theory, we prove the main statements in Notaris (Numer. Math.,142:129-147,2019) in a different manner, by means of the Jacobi tridiagonal matrix approach.