By Jorge Bustamante

This publication comprises an exposition of numerous effects similar with direct and speak theorems within the idea of approximation by means of algebraic polynomials in a finite period. additionally, a few proof relating trigonometric approximation which are important for motivation and comparisons are integrated. the choice of papers which are referenced and mentioned rfile a few tendencies in polynomial approximation from the Nineteen Fifties to the current day.

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**Extra info for Algebraic Approximation: A Guide to Past and Current Solutions**

**Example text**

As Shevchuk showed, the converse of the last result is not true. 11 (Shevchuk, [338]). 26). / W r Hk [ψ]. – There exists a function f for which En (f ) = O(n−2r ψ(n2 )) and f ∈ r – There exists a function f ∈ / W Hk [ϕ] and a sequence {Pn } of polynomials such that the Timan estimate holds. 11 for k = 1, r = 0) was obtained earlier by Dolzhenko and Sevastyanov [103]. 34 Chapter 2. 12 (Shevchuk, [338]). For any function ϕ ∈ Φk , there is a function f ∈ W r Hkϕ such that (i) For all n ∈ N , En (f ) ≤ n−2r ϕ(n−2 ), (ii) ωk (f (r) , t) ≥ cϕ(t), t ∈ [0, 1/k], c = c(r, k) > 0.

That is, we have a characterization of functions satisfying a classical Lipschitz condition in terms of the rate of pointwise approximation by algebraic polynomials. Let us consider the problem of characterization of other classes of functions. For r ∈ N and α ∈ (0, 1), let K(r, α) = {f ∈ C[−1, 1] : En (f ) ≤ M (f )n−r−α }. Classes K(r, α) are deﬁned in terms of the rate of convergence of the best approximation. The classes C r,α [−1, 1] and K(r, α) are diﬀerent. For instance, for √ f (x) = 1 − x2 one has, f ∈ K(0, 1) but, for any δ > 1/2, f ∈ / C 0,δ [−1, 1].

If | Pn (x) | ≤ K [Δn (x)]−k ψ(n) where ψ(n) is decreasing, ψ(n) = o(1), and satisﬁes some additional conditions, then ωrϕ (f, 1/n) ≤ M ψ(n). This provides the analogue to the Sunouchi-Zamanski theorem. 5 (Ditzian, [95]). If for some integer r and decreasing sequence ψ(n), l 2kr ψ(2k ) ≤ M 2lr ψ(2l ) and En (f ) ≤ ψ(n), k=1 then for Pn , the polynomial satisfying f − Pn = En (f ), one has | Pn(k) (x) | ≤ K [Δn (x)]−k ψ(n). In particular, if for some r, l 2kr E2k (f ) ≤ M 2lr E2l (f ) k=1 then | Pn(k) (x) | ≤ K [Δn (x)]−k En (f ).