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On bounds for the Lipschitz constant of the remainder in polynomial approximation

Published online by Cambridge University Press:  17 April 2009

David Elliott
David Elliott
Affiliation:
Department of Mathematics, University of Tasmania, Box 252C, G.P.O., Hobart, Tasmania, Australia.
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Abstract

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Suppose f is a function possessing a kth order derivative, the derivative being Lipschitz continuous of order α, 0 < α ≤ 1, on [−1, 1]. Let Pn be a polynomial of degree ≤ n approximating to f on [−1,1] such that if rn = f − Pn then ∥rn∥∞ ≤ An−k−a Define

where 0 < β ≤ 1. Upper bounds are obtained for Mn(β) when k ≥ 1 thereby generalizing results previously given for functions which are only Lipschitz continuous on [−1,1].

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 33 , Issue 1 , February 1986 , pp. 49 - 57
Copyright
Copyright © Australian Mathematical Society 1986

References

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