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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

[1]Ioakimidis, N.I., “An improvement of Kalandiya's theorem”, J. Approx. Theory 38 (1983), 354–356.CrossRefGoogle Scholar
[2]Kalandiya, A.I., “On a direct method of solution of an equation in wing theory and its application to the theory of elasticity”, Mat. Sb. 42 (1957), 249–272. (In Russian) Google Scholar
[3]Meinardus, G., Approximation of functions: theory and nwnerical methods, (Springer–verlag, Berlin, Heidelberg, New York, 1967). CrossRefGoogle Scholar
[4]Stečkin, S.B., “Generalization of some Bernstein inequalities”, Dok l. Akad. Nauk SSSR 60 (1948), 1511–1514. (In Russian).Google Scholar