Sharp error bounds for Jacobi expansions and Gegenbauer-Gauss quadrature of analytic functions (Q2845598)

The authors claim sharp bounds for the Jacobi expansion coefficients of analytic functions defined on the closure of a Bernstein ellipse. For these coefficients they use a classical representation in numerical series involving products on an integral along the ellipse contour and some scalar product-like coefficients. These scalar product coefficients come from the usual scalar product of Jacobi and Chebyshev polynomials of the second kind with respect to the Jacobi weight function. The main results (Lemma 2.4 and Theorem 2.5) provide a bound for the Jacobi expansion coefficients which contains in its right hand side a series of absolute values of successive differences of scalar products coefficients. The sharpness of the bounds of the Jacobi expansion coefficients depends on the rate of convergence of this series. However, the asymptotic bounds provided by the authors do not show a considerable improvement with respect to the most recent ones (Remark 2.5). Some error estimates are also provided for the Gegenbauer-Gauss quadrature rule.