By Jorge Bustamante

This ebook comprises an exposition of a number of effects comparable with direct and speak theorems within the conception of approximation through algebraic polynomials in a finite period. moreover, a few evidence relating trigonometric approximation which are worthwhile for motivation and comparisons are integrated. the choice of papers which are referenced and mentioned record a few traits in polynomial approximation from the Fifties to the current day.

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22) and let f : [−1, 1] → R be a function. 21). In order to obtain the converse result, diﬀerent variants of the Bernstein inequality are needed. That is we should estimate the derivatives of an algebraic polynomial in terms of the polynomial. 4. Assume that r, n ∈ N and let [−1, 1]. · denote the uniform norm on (i) (Markov, [248]) If Pn ∈ Pn , then Pn(r) ≤ n2r Pn . 7. Characterization of some classes of functions 29 (ii) (Bernstein, [27]) If Pn ∈ Pn and x ∈ [−1, 1], then 1 − x2 Pn (x) ≤ n Pn .

I) (Dzyadyk 1956, [107]) If for x ∈ [−1, 1] a polynomial Pn ∈ Pn satisﬁes the inequality 1 | Pn (x) |≤ L ( 1 − x2 )ρ + ρ , n then there exists a constant C (which depends only on ρ) such that, for x ∈ (−1, 1) one has | Pn (x) |≤ C n L min ( 1 − x2 )ρ−1 , 1 nρ−1 , if ρ≤1 and | Pn (x) |≤ C n L ( 1 − x2 )ρ−1 + 1 nρ−1 , if ρ ≥ 1. (ii) (Potapov 1960, [288]) If ρ, γ ∈ R, there exists a constant C such that, if for x ∈ [−1, 1] a polynomial Pn ∈ Pn satisﬁes the inequality ρ | Pn (x) |≤ L (n + 1)γ+ρ (Δn+1 (x)) , then for x ∈ (−1, 1) one has | Pn (x) |≤ C L (n + 1)γ+ρ (Δn+1 (x))ρ−1 .

It can be proved that these functions are continuous. Moreover, as in the proof of the case r = 0, each function Ψi satisﬁes a Lipschitz condition of order α. Therefore, there exist a constant C and a sequence {Tn } of even trigonometric polynomials such that | F (t) − Tn (t) |≤ Cn−(k+α) . By taking Pn (x) = Tn (arccos x) we conclude that f ∈ K(r, α). Let us consider the relation K(r, α) ⊂ S(r, α). Fix f ∈ K(r, α) and a sequence {Pn } of polynomials such that f − Pn ≤ Cn−(k+α) . 45) 48 Chapter 2.