Comparison theorems for compound quadrature formulas (Q1092350)

The aim of the paper is to compare the rates of convergence of quadrature processes. The author gives some representation formulas for the remainders of one process in terms of a second one. For instance, let \(R_ n^{tr}(f)\), \(R_ n^{Si}(f)\), \(R_ n^{mi}(f)\) denote the remainders in the (n-fold compound) trapezoidal, Simpson, and midpoint rule, respectively, Let R[0,1] be the linear space of all Riemann integrable functions on [0,1]. If f belongs to a certain subspace of R[0,1], then it is shown that \(R_ n^{tr}(f)=- 3\sum^{\infty}_{k=0}4^ k\cdot R^{Si}_{n2^ k}(f).\) The following results are also established: (I) Let \(\delta >1\), \(f\in R[0,1]\) and \(R_ n^{tr}(f)=o(n^{-1})\) for \(n\to \infty\). Then (*) \(R_ n^{tr}(f)=O(n^{-\delta})\), \(R_ n^{mi}(f)=O(n^{-\delta})\), \(n\to \infty\). (II) When \(\delta\in [0,1)\) and \(f\in R[0,1]\), (*) is true for the subsequence \(n=2^ m\), \(m\to \infty\). In the proofs, the main tool is the Möbius inversion.