Interpolatory Pointwise Estimates for Polynomial Approximation

We discuss whether or not it is possible to have interpolatory pointwise estimates in the approximation of a function f 2 C 0; 1], by polynomials. For the sake of completeness as well as in order to strengthen some existing results, we discuss brieey the situation in unconstrained approximation. Then we deal with positive and monotone constraints where we show exactly when such interpolatory estimates are achievable by proving aarmative results and by providing the necessary counterexamples in all other cases. The eeect of the endpoints of the nite interval on the quality of approximation of continuous functions by algebraic polynomials, was rst observed by Nikolski Nik46]. Later pointwise estimates of this phenomenon were given by Timan Tim51] (k = 1), Dzjadyk Dzj58, Dzj77] (k = 2), Freud Fre59] (k = 2), and Brudny Bru63] (k 2), who proved that if f 2 C r 0; 1], then for each n N = r + k ? 1, a polynomial p n 2 n exists, such that (1.1) jf(x) ? p n (x)j c(r; k) r n (x)! k (f (r) ; n (x)); 0 x 1: where n (x) := 1 n 2 + 1 n '(x), '(x) := p x(1 ? x), and where ! k , k 1, is the ordinary kth modulus of smoothness, and c(r; k) is a constant which depends only on r and k, and which is independent of f and n. Note that for n = r + k ? 1, (1.1) is in fact a Whitney inequality Whi57]. Lorentz at a conference in Oberwolfach Lor64], raised the question whether n (x) can be replaced by the smaller quantity n (x) := 1 n '(x), that is, whether (1.2) jf(x) ? p n (x)j c(r; k) r n (x)! k (f (r) ; n (x)); 0 x 1; holds. Such estimates we call interpolatory pointwise estimates. 1 The authors acknowledge partial support for this work by the Volkswagen{Stiftung (RiP program at Oberwolfach). The third author's contribution was also sponsored under the program \FF orderung der Europaff ahigkeit der Hochschulen" of the state of Nordrhein{Westfalen.