Mathematical Analysis

# Read e-book online Algebraic Approximation: A Guide to Past and Current PDF

By Jorge Bustamante

ISBN-10: 3034801939

ISBN-13: 9783034801935

This publication comprises an exposition of numerous effects similar with direct and speak theorems within the conception of approximation via algebraic polynomials in a finite period. moreover, a few proof referring to trigonometric approximation which are beneficial 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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Additional info for Algebraic Approximation: A Guide to Past and Current Solutions

Sample text

If f is not a polynomial, there exist constants M (f, k), such that Pn(k) (f ) ≤ M (f, r) n2k−r ω f (r) , 1 n n ≥ 1. Proof. The proof of this theorem is based on an extension of f . Fix two reals c and d (c < a and b > d) and assume that f has been extended to a function F ∈ C r [c, d] in such a way that ω(F (r) , h) ≤ Cω(f (r) , h). 38 Chapter 2. The End Points Eﬀect Fix a sequence {Qn } of polynomials such that (k) Q(k) n −F [c,d] ≤ K nr−k ω F (r) , (k) for k ≤ r and n ≥ k + 1. One has Qn (k) Qn [c,d] Bernstein’s inequality) Pn(k) [a,b] ≤ Kk n 1 n ≤ [c,d] CK 1 ω f (r) , r−k n n , ≤ Ck , for 0 ≤ k ≤ r and (by , for k > r.

40) where the constants C(m, k) depend only on m and k. An algebraic analog of the result of Czipszer and Freud in [78] is the following. 18 (Kilgore and Szabados, [196]). Let g ∈ C q [−1, 1] be such that g (k) (±1) = 0 for k ≤ q − 1. Let ε > 0 and assume there is a sequence {Pn+q } (Pn+q ∈ Pn+q ) such that g(x) − Pn+q (x) ε √ ≤ q. 2 q n ( 1−x ) Then, for |x| ≤ 1 and k ≤ q, g(x) − pn (x) (k) ≤ √ 1 − x2 1 + 2 n n where δk,q and γk,q ε depend on k and q. q−k δk,q inf (g − pn )(q) + γk,q ε , pn 44 Chapter 2.

R+α) The characterization of the class Hp 1971. He veriﬁed that the quantity λn (f ) = inf P ∈P was also considered by Motornyi in f (x) − P (x) (Δn (x))α Lp (α) (r+α) (r+α) are unbounded in the class Hp and established that classes Hp and Ap are diﬀerent for 0 < α < 1 and coincide for α = 1. He also characterized some functions, but not in terms of approximation by polynomials (see Theorem 11 of [255]) on the whole interval. Oswald [277] extended some of the results of Motornyi to the case of moduli of smoothness of higher order.