A $K$-functional and the rate of convergence of some linear polynomial operators
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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 (\Vert {f-g}+n^{-2} \Vert {P(D) g} \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.References
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Additional Information
- Z. Ditzian
- Affiliation: Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
- MR Author ID: 58415
- 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
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 1773-1781
- MSC (1991): Primary 41A10, 41A35, 41A25
- DOI: https://doi.org/10.1090/S0002-9939-96-03219-4
- MathSciNet review: 1307511