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A $K$-functional and the rate of convergence
of some linear polynomial operators

Author: Z. Ditzian
Journal: Proc. Amer. Math. Soc. 124 (1996), 1773-1781
MSC (1991): Primary 41A10, 41A35, 41A25
MathSciNet review: 1307511
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Abstract: We show that the $K$-functional

\begin{equation*}K(f,n^{-2} )_{p}=\inf _{g\in C^{2}[-1,1]} \bigl (\|f-g\|_p+n^{-2} \|P(D) g\|_p \bigr ), \end{equation*}

where $P(D) =\frac {d}{dx} (1-x^{2})\frac {d}{dx} $, is equivalent to the rate of convergence of a certain linear polynomial operator. This operator stems from a Riesz-type summability process of expansion by Legendre polynomials. We use the operator above to obtain a linear polynomial approximation operator with a rate comparable to that of the best polynomial approximation.

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Additional Information

Z. Ditzian
Affiliation: Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

Keywords: Linear polynomial approximation, near best polynomial approximation
Received by editor(s): April 6, 1994
Received by editor(s) in revised form: November 18, 1994
Additional Notes: Supported by NSERC grant A4816 of Canada.
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1996 American Mathematical Society