On a quadrature formula of Gori and Micchelli

摘要

Sparked by Bojanov (J. Comput. Appl. Math. 70 (1996) 349), we provide an alternate approach to quadrature formulas based on the zeros of the Chebyshev polynomial of the first kind for any weight function w introduced and studied in Gori and Micchelli (Math. Comp. 65 (1996) 1567), thereby improving on their observations. Upon expansion of the divided differences, we obtain explicit expressions for the corresponding Cotes coefficients in Gauss–Turán quadrature formulas for I(f;w)≔∫-11f(x)w(x)dx and I(fTn;w) for a Gori–Micchelli weight function. It is also interesting to mention what has been neglected for about 30 years by the literature is that, as a consequence of expansion of the divided differences in the special case when w(x)=1/1-x2, the solution of the famous Turán's Problem 26 raised in 1980 was in fact implied by a result of Micchelli and Rivlin (IBM J. Res. Develop. 16 (1972) 372) in 1972. Some concluding comments are made in the final section.