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Progress in Approximation Theory and Applicable Complex Analysis pp 463–489Cite as

A New Approach to Positivity and Monotonicity for the Trapezoidal Method and Related Quadrature Methods

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Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 117))

Abstract

For positive integers n let R n [f] be the remainders of a quadrature method applied to a function f. It is of practical importance to know sufficient conditions on f which guarantee that the remainders are non-negative and converge monotonically to zero as n → ∞. For most of the familiar quadrature methods such conditions are known as sign conditions on certain derivatives of f. However, conditions of this type specify only a small subset of the desired functions. In particular, they exclude oscillating functions. In the case of the trapezoidal method, we propose a new approach based on Fourier analysis and the theory of positive definite functions. It allows us to describe much wider classes of functions for which positivity and monotonicity occur. Our considerations include not only the trapezoidal method on a compact interval but also that for integration over the whole real line as well as some related methods.

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Authors and Affiliations

  1. Département de Mathématiques et de Statistique, Université de Montréal, Montréal, QC, Canada, H3T 1J4

    Q. I. Rahman

  2. Department of Mathematics, University of Erlangen-Nuremberg, Cauerstr. 11, 91058, Erlangen, Germany

    G. Schmeisser

Authors
  1. Q. I. Rahman
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  2. G. Schmeisser
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Correspondence to G. Schmeisser .

Editor information

Editors and Affiliations

  1. Department of Mathematics and Statistics, Auburn University, Auburn, Alabama, USA

    Narendra Kumar Govil

  2. Department of Mathematics, University of Central Florida, Orlando, Florida, USA

    Ram Mohapatra

  3. Department of Mathematics, Tuskegee University, Tuskegee, Alabama, USA

    Mohammed A. Qazi

  4. Department of Mathematics, University of Erlangen-Nuremberg, Erlangen, Germany

    Gerhard Schmeisser

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Rahman, Q.I., Schmeisser, G. (2017). A New Approach to Positivity and Monotonicity for the Trapezoidal Method and Related Quadrature Methods. In: Govil, N., Mohapatra, R., Qazi, M., Schmeisser, G. (eds) Progress in Approximation Theory and Applicable Complex Analysis. Springer Optimization and Its Applications, vol 117. Springer, Cham. https://doi.org/10.1007/978-3-319-49242-1_21

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