Eine wahrscheinlichkeitstheoretische Begründung der Integrationsformeln von Newton-Cotes

Summary

A definite integral is approximated by a linear combination of values of the integrand at equidistant points. As a measure for the goodness of fit we use the mean square deviation; for this we take a wide sense stationary stochastic process as basis. We prove that there exist always a best integration formula in this sense. The Taylor expansion of its coefficients begins with the coefficients ofNewton-Cotes. It is shown that the error of the best integration formula is of the same order of magnitude as in the well-known formula ofNewton-Cotes. Therefore we get a justification for the formulae ofNewton-Cotes also if the integrand is not differentiable. Besides we obtain estimates of the error, which use higher differences instead of higher differential quotients of the integrand.