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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Monotone and comonotone approximation


Authors: E. Passow and L. Raymon
Journal: Proc. Amer. Math. Soc. 42 (1974), 390-394
MSC: Primary 41A50
MathSciNet review: 0336176
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Abstract: Jackson type theorems are obtained for monotone and comonotone approximation. Namely

(i) If $ f(x)$ is a function such that the kth difference of f is $ \geqq 0$ on [a, b] then the degree of approximation of f by nth degree polynomials with kth derivative $ \geqq 0$ is $ 0[\omega (f;1/{n^{1 - \varepsilon }})]$ for any $ \varepsilon > 0$, where $ \omega (f;\delta )$ is the modulus of continuity of f on [a, b].

(ii) If $ f(x)$ is piecewise monotone on [a, b] then the degree of approximation of f by nth degree polynomials comonotone with f is $ 0[\omega (f;1/{n^{1 - \varepsilon }})]$ for any $ \varepsilon > 0$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1974-0336176-9
PII: S 0002-9939(1974)0336176-9
Keywords: Monotone approximation, comonotone approximation, piecewise monotone approximation, Jackson theorem
Article copyright: © Copyright 1974 American Mathematical Society