Extended Euler-Maclaurin summation formulas
The well-known Euler-Maclaurin summation formula may be viewed as an asymptotic formula for the sum \((1/n) \sum_{k=0}^{n-1} f((k+\theta)/n)\), \(0<\theta \leq 1\), for \(n\) tending to infinity. Usually, the function \(f\) is assumed to have continuous derivatives up to some order on the interval \([0,1]\). The paper, which is a revision and extension of earlier papers of Navot and the author, gives asymptotics in the case of \(f\) having some singularity at one endpoint of the interval (of the form \(x^{-\alpha}\) \(\alpha>0\), where \(\alpha \not\in \mathbb{N}\) or \(x^{-\alpha} \log x\) where \(\alpha\geq 0\), \(\alpha\not\in \mathbb{N}\)). The idea is the following: \(f\) is represented in a form \(f= f_ 1+h\) where the function \(h\) is smooth and \(f_ 1\) is a finite sum of elementary functions having singularities. Then the Euler-Maclaurin formula is applied to the functions \(f_ 1\) and \(h\) separately.
scientific article; zbMATH DE number 578649