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Open Access Research Article

Approximating the Riemann-Stieltjes integral of smooth integrands and of bounded variation integrators

SS Dragomir12* and S Abelman1

Author Affiliations

1 School of Computational & Applied Mathematics, University of the Witwatersrand, Private Bag-3, Wits-2050, Johannesburg, South Africa

2 Mathematics, College of Engineering & Science, Victoria University, P.O. Box 14428, Melbourne, MC 8001, Australia

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Journal of Inequalities and Applications 2013, 2013:154  doi:10.1186/1029-242X-2013-154

Published: 4 April 2013

Abstract

In the present paper, we investigate the problem of approximating the Riemann-Stieltjes integral <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M1">View MathML</a> in the case when the integrand f is n-time differentiable and the derivative <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M2">View MathML</a> is either of locally bounded variation, or Lipschitzian on an interval incorporating <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/154/mathml/M3">View MathML</a>. A priory error bounds for several classes of integrators u and applications in approximating the finite Laplace-Stieltjes transform and the finite Fourier-Stieltjes sine and cosine transforms are provided as well.

MSC: 41A51, 26D15, 26D10.

Keywords:
Riemann-Stieltjes integral; Taylor’s representation; functions of bounded variation; Lipschitzian functions; integral transforms; finite Laplace-Stieltjes transform; finite Fourier-Stieltjes sine and cosine transforms