Error bounds of Micchelli-Rivlin quadrature formula for analytic functions

Abstract

We consider the well known Micchelli-Rivlin quadrature formula, of highest algebraic degree of precision, for the Fourier-Chebyshev coefficients. For analytic functions the remainder term of this quadrature formula can be represented as a contour integral with a complex kernel. We study the kernel, on elliptic contours with foci at the points -/+ 1 and a sum of semi-axes rho > 1, for the quoted quadrature formula. Starting from the explicit expression of the kernel, we determine the locations on the ellipses where maximum modulus of the kernel is attained. So we derive effective L-infinity-error bounds for this quadrature formula. Complex-variable methods are used to obtain expansions of the error in the Micchelli-Rivlin quadrature formula over the interval [-1, 1]. Finally, effective L-1-error bounds are also derived for this quadrature formula.