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

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 ).

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