IDEAS home Printed from https://ideas.repec.org/h/spr/spochp/978-3-319-49242-1_21.html
   My bibliography  Save this book chapter

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

In: Progress in Approximation Theory and Applicable Complex Analysis

Author

Listed:
  • Q. I. Rahman

    (Université de Montréal)

  • G. Schmeisser

    (University of Erlangen-Nuremberg)

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.

Suggested Citation

  • Q. I. Rahman & G. Schmeisser, 2017. "A New Approach to Positivity and Monotonicity for the Trapezoidal Method and Related Quadrature Methods," Springer Optimization and Its Applications, in: Narendra Kumar Govil & Ram Mohapatra & Mohammed A. Qazi & Gerhard Schmeisser (ed.), Progress in Approximation Theory and Applicable Complex Analysis, pages 463-489, Springer.
    • Handle: RePEc:spr:spochp:978-3-319-49242-1_21
    DOI: 10.1007/978-3-319-49242-1_21
    as

    Download full text from publisher

    To our knowledge, this item is not available for download. To find whether it is available, there are three options:
    1. Check below whether another version of this item is available online.
    2. Check on the provider's web page whether it is in fact available.
    3. Perform a search for a similarly titled item that would be available.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:spochp:978-3-319-49242-1_21. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.