The Iyengar inequality,revisited

Detailed analysis shows that the famous Iyengar inequality actually says that the Trapezoidal formula is a central algorithm for approximating integrals over an appropriate interval for the class of functions whose derivatives are bounded by a positive number K in L _∞-sense. The inherent nonlinearity from central algorithms reflects the importance of the Iyengar inequality and thus makes familiar linear methods malfunction when one tries to generalize it. It is shown that the generalization depends on a nonlinear system of equations satisfied by a set of free nodes of a perfect spline. Explicit constructions are obtained in the spirit of the Iyengar inequality for the class of functions whose rth (r≤4) derivatives are bounded by a positive number K in L _∞-sense because a closed solution to the nonlinear system is only available for r≤4. Connections with computational mathematics, especially with best interpolation and best quadrature, are discussed. Numerical experiments are also included. AMS subject classification (2000)  65D30, 41A17