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We show that the K-functional K(f, n−2)p = inf g∈C2[−1,1] ( ‖f − g‖p + n−2‖P (D)g‖p ) , where P (D) = d dx (1 − x2) 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.

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