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Degree of Approximation by Rational Functions with Prescribed Numerator Degree

Published online by Cambridge University Press:  20 November 2018

D. Leviatan  and
D. S. Lubinsky
D. Leviatan
Affiliation:
Department of Mathematics Sackler Faculty of Science Tel Aviv University Ramat Aviv, Tel Aviv Israel
D. S. Lubinsky
Affiliation:
Department of Mathematics Witwatersrand University P.O. Wits 2050 South Africa
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Abstract

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We prove a Jackson type theorem for rational functions with prescribed numerator degree: For continuous functions f: [—1,1] —> ℝ with ℓ sign changes in (—1,1), there exists a real rational function Rℓ,n(x) with numerator degree ℓ and denominator degree at most n, that changes sign exactly where f does, and such that

Here C is independent of f, n and ℓ, and ωφ is the Ditzian-Totik modulus of continuity. For special functions such as f(x) = sign(x)|x|α we consider improvements of the Jackson rate.

Keywords

41A2041A25rational approximationreal rational approximationJackson theoremsmoduli of continuityprescribed numerator degree
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 46 , Issue 3 , 01 June 1994 , pp. 619 - 633
Copyright
Copyright © Canadian Mathematical Society 1994

References

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