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Optimal quadrature formulae and minimal monosplines in Lq

Published online by Cambridge University Press:  09 April 2009

J. Kautsky
J. Kautsky
Affiliation:
Cf Discipline of Applied MathematicsThe Flinders University of South Australia
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Summary

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The quadrature formula of order m using values of derivatives up to the m — 1st order with the best possible bound in is derived. Using certain properties of the polynomials minimal in Lq norm, it is proved that the optimal formula not use the derivatives of m — 1st order if m is even.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 11 , Issue 1 , February 1970 , pp. 48 - 56
Copyright
Copyright © Australian Mathematical Society 1970

References

[1]Krylov, V. I., Approximate Calculations of Integrals (MacMillan, New York, 1962, Translated from Russian).Google Scholar
[2]Schoenberg, I. J., ‘On Interpolation by Spline Functions and its Minimal Properties’, On Approximation Theory, Proc. of the Conference, 08 4—10, 1963, ISNM, Vol. 5.Google Scholar
[3]Timan, A. F., Theory of Approximations of Functions of a Real Variable, (Pergamon Press, 1963, Translated from Russian).Google Scholar