Modified compound quadrature rules for strongly singular integrals

We show that the error term of every modified compound quadrature rule for Cauchy principal value integrals with degree of exactnesss is of optimal order of magnitude in the classesCk[−1,1],k=1,2,...,s, but not inCs+1[−1,1]. We give explicit upper bounds for the error constants of the modified midpoint rule, the modified trapezoidal rule and the modified Simpson rule. Furthermore, the results are generalized to analogous rules for Hadamard-type finite part integrals.ZusammenfassungWir zeigen, daß der Fehlerterm jedes modifizierten zusammengesetzten Quadraturverfahrens für Cauchy-Hauptwert-Integrale mit Exaktheitsgrads die bestmögliche Größenordnung in den KlassenCk[−1,1],k=1,2,...,s hat, aber nich inCs+1[−1,1]. Explizite obere Schranken für die Fehlerkonstanten der modifizierten Mittelpunkt-, Trapez- und Simpson-Verfahren werden angegeben. Des weiteren werden die Ergebnisse auf die entsprechenden Verfahren für Finite-Part-Integrale vom Hadamardschen Typ verallgemeinert.

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