Integral formulas for Chebyshev polynomials and the error term of interpolatory quadrature formulae for analytic functions

Abstract:

We evaluate explicitly the integrals $\int _{-1}^{1}\pi _{n}(t)/(r\mp t)dt,\ |r|\neq 1$, with the $\pi _{n}$ being any one of the four Chebyshev polynomials of degree $n$. These integrals are subsequently used in order to obtain error bounds for interpolatory quadrature formulae with Chebyshev abscissae, when the function to be integrated is analytic in a domain containing $[-1,1]$ in its interior.