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Gauss-Kronrod quadrature error estimates for analytic functions. (English) Zbl 0806.41018
Error bounds are proved for the Gauss-Kronrod quadrature formula and analytic functions, using the maximum modulus of the integrand along closed contours in the complex plane, and in particular along circles and ellipses.
Furthermore, the remainder functional is investigated for the first $$\nu$$ (for increasing polynomial degree) Chebyshev polynomials of the first and second kind which are not integrated exactly. It is shown, for sufficiently large $$n$$ and independent $$\nu$$, that these values are negative, and that their absolute values increase with the polynomial degree.

##### MSC:
 41A55 Approximate quadratures 41A80 Remainders in approximation formulas 30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 65D32 Numerical quadrature and cubature formulas