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The kernel of a rule of approximate integration

Published online by Cambridge University Press:  17 February 2009

J. H. Loxton and
J. W. Sanders
J. H. Loxton
Affiliation:
School of Mathematics, University of New South Wales, Kensington, N.S.W. 2033
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Abstract

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It is well known that the trapezoidal rule of quadrature is exact for linear functions on [0, 1], and easy to see that it is exact for functions of the form f = l+g where l is linear and g is odd about ½. Not so well known is an example of a continuous function for which the trapezoidal rule is exact but which does not have this form. In this paper we show that if the trapezoidal rule is exact for f then f has the form above provided it has either absolutely convergent Fourier series or continuous second derivative. We consider one-sided versions in which the approximate integrals are non-negative, and also characterize those sequences arising as the approximate integrals of a function with absolutely convergent Fourier series.

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 21 , Issue 3 , January 1980, pp. 257 - 267
Copyright
Copyright © Australian Mathematical Society 1980

References

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