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ISSN 1088-6826(e) ISSN 0002-9939(p)

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

Author(s): 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.

References:

1.: R. Askey, Orthogonal polynomials and special functions, Regional Conference Series in Applied Mathematics, vol. 21, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1975. MR 58:1288
2.: R. Askey and I.I. Hirschman, Mean summability for ultraspherical polynomials, Math. Scand. 12 (1963), 167--177. MR 29:1497
3.: W. Chen and Z. Ditzian, Strong converse inequality for Kantorovich polynomials, Const. Approx. 10 (1994), 95--106. MR 94k:41039
4.: W. Chen, Z. Ditzian, and K. Ivanov, Strong converse inequality for the Bernstein-Durrmeyer operator, J. Approx. Theory 75 (1993), 25--43. MR 94h:41047
5.: M. Derrienic, Sur l'approximation de fonctions intégrales sur par des polynômes de Bernstein modifies, J. Approx. Theory 31 (1981), 325--343.
6.: Z. Ditzian and V. Totik, Moduli of smoothness, Springer-Verlag, 1987. MR 89h:41002
7.: Z. Ditzian and K. Ivanov, Strong converse inequalities, J. d'Analyse Math. 61 (1993), 61--111. MR 94m:41038
8.: H. Pollard, The mean convergence of orthogonal series I, Trans. Amer. Math. Soc. 62 (1947), 387--403. MR 9:280d
9.: V. Totik, Approximation by Bernstein polynomials, Amer. J. Math. 116 (1994), 995--1018. CMP 94:16

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Additional Information:

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

DOI: 10.1090/S0002-9939-96-03219-4
PII: S 0002-9939(96)03219-4
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
Copyright of article: Copyright 1996, American Mathematical Society

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