Optimale Quadraturformeln mit semidefiniten Peano-Kernen

Summary The remainder in numerical integration formulas can be represented asR(f)=cf ^(m) (), if and only if the associated Peano kernel is semidefinite. We investigate the question of optimal constantsc. Our existence theorems generalize a result of Schmeisser 12]. In addition we prove characterizing statements of optimal formulas. In particular we show: i) The Peano kernels have a maximal number of zeros. ii) The weights are positive and the inner knots have (implicitly) order two. iii) The formulas are of interpolatory type.