Download Algebraic Approximation: A Guide to Past and Current by Jorge Bustamante PDF

By Jorge Bustamante

This publication includes an exposition of a number of effects comparable with direct and speak theorems within the idea of approximation via algebraic polynomials in a finite period. moreover, a few evidence touching on trigonometric approximation which are worthy for motivation and comparisons are incorporated. the choice of papers which are referenced and mentioned rfile a few traits in polynomial approximation from the Fifties to the current day.

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In 1980 Potapov [295] unified the results of Dzyadyk and Fuksman. 3, but with the condition α + β < 1 instead of α + β/2 < 1. Notice that, by taking β = 0 we obtain the Dzyadyk characterization and for β = −α the Fuksman result. The results we present here were proved by Potapov in 2005 [303]. 3 (Potapov, [303]). Fix reals α and β such that α ∈ (0, 1), α + β ≥ 0 and α + β/2 < 1. For f ∈ C[−1, 1] the following assertions are equivalent: (i) For each x ∈ [−1, 1], one has sup { h : |h|≤δ, |x+h|≤1} | f (x + h) − f (x) |≤ C1 δ α 1 − x2 + √ δ β , where C1 is a positive constant which does not depend on δ or x; (ii) For each n ∈ N there exists Pn−1 ∈ Πn−1 such that, for each x ∈ [−1, 1], | f (x) − Pn−1 (x) | ≤ C2 nβ (Δn (x)) α+β where C2 is a positive constant which does not depend on x or n.

The following assertions are equivalent: i) f ∈ C r [−1, 1] and for each x ∈ [−1, 1], sup {h: |h|≤δ, |x+h|≤1} | f (r) (x) − f (r) (x + h) |≤ C δ, where C is a positive constant which does not depend on x or δ; ii) For each n ∈ N there exists Pn ∈ Pn such that, for each x ∈ [−1, 1], | f (x) − Pn (x) | ≤ D nr+α 1 − x2 + 1 n r+α , where D is a positive constant which does not depend on x or δ. 28 Chapter 2. The End Points Effect In 1957 Timan presented a converse result assuming that an estimate in terms of the function Δn (x) is known.

Is it possible to obtain a speed of approximating at the ends of the segment greater than the one in Timan’s theorem? He considered that the rate of approximation on the whole segment is retained. Both questions were answered in the negative. For instance, he proved that an estimation of the form | f (x) − pn (x) | ≤ Cω f, ψ1 (| x − a |) + ψ2 (1/n) n , is not possible for all f ∈ C[−1, 1], where | a |< 1 and ψi is an increasing function satisfying ψi (t) → 0 as t → 0 (i = 1, 2). Moreover, we can not replace the expression Δn (x) by o(Δn (x)).

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