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

Ditzian, Z.

doi:10.1090/S0002-9939-96-03219-4

A $K$-functional and the rate of convergence of some linear polynomial operators
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Proc. Amer. Math. Soc. 124 (1996), 1773-1781 Request permission

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.

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