TY - JOUR

T1 - Uniform and pointwise shape preserving approximation (SPA) by algebraic polynomials

T2 - an update

AU - Kopotun, K. A.

AU - Leviatan, D.

AU - Shevchuk, I. A.

N1 - Publisher Copyright:
© Société de Mathématiques Appliquées et Industrielles, 2019 Certains droits réservés.

PY - 2019

Y1 - 2019

N2 - It is not surprising that one should expect that the degree of constrained (shape preserving) approximation be worse than the degree of unconstrained approximation. However, it turns out that, in certain cases, these degrees are the same. The main purpose of this paper is to provide an update to our 2011 survey paper. In particular, we discuss recent uniform estimates in comonotone approximation, mention recent developments and state several open problems in the (co)convex case, and reiterate that co-q-monotone approximation with q ≥ 3 is completely different from comonotone and coconvex cases. Additionally, we show that, for each function f from ∆(1), the set of all monotone functions on [-1, 1], and every α > 0, we have nα(f - Pn) lim sup nα(f - Pn) inf ≤ c(α) lim sup inf ϕα n→∞ Pn∈Pn∩∆(1) ϕα Pn∈Pn n→∞ where Pn denotes the set of algebraic polynomials of degree < n, ϕ(x):= √1 - x2, and c = c(α) depends only on α.

AB - It is not surprising that one should expect that the degree of constrained (shape preserving) approximation be worse than the degree of unconstrained approximation. However, it turns out that, in certain cases, these degrees are the same. The main purpose of this paper is to provide an update to our 2011 survey paper. In particular, we discuss recent uniform estimates in comonotone approximation, mention recent developments and state several open problems in the (co)convex case, and reiterate that co-q-monotone approximation with q ≥ 3 is completely different from comonotone and coconvex cases. Additionally, we show that, for each function f from ∆(1), the set of all monotone functions on [-1, 1], and every α > 0, we have nα(f - Pn) lim sup nα(f - Pn) inf ≤ c(α) lim sup inf ϕα n→∞ Pn∈Pn∩∆(1) ϕα Pn∈Pn n→∞ where Pn denotes the set of algebraic polynomials of degree < n, ϕ(x):= √1 - x2, and c = c(α) depends only on α.

KW - Approximation by algebraic polynomials

KW - constrained approximation

KW - shape preserving approximation

UR - http://www.scopus.com/inward/record.url?scp=85128850904&partnerID=8YFLogxK

U2 - 10.5802/smai-jcm.54

DO - 10.5802/smai-jcm.54

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AN - SCOPUS:85128850904

SN - 2426-8399

VL - S5

SP - 99

EP - 108

JO - SMAI Journal of Computational Mathematics

JF - SMAI Journal of Computational Mathematics

ER -