Interpolatory pointwise estimates for convex polynomial approximation

2020/01/11 by K. A. Kopotun, D. Leviatan, Kopotun, K. A. +5
Mathematics · #41A05 #41A10 #41A25 #41A29 #Advanced Banach Space Theory #Approximation Theory and Sequence Spaces #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2001.03769

openalex publication_date 2020/01/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper deals with approximation of smooth convex functions f on an interval by convex algebraic polynomials which interpolate f at the endpoints of this interval. We call such estimates "interpolatory". One important corollary of our main theorem is the following result on approximation of f∈ Δ⁽²⁾, the set of convex functions, from W^r, the space of functions on [-1,1] for which f^(r-1) is absolutely continuous and ‖f^(r)‖_∞ := ess sup_x∈[-1,1] |f^(r)(x)| < ∞: For any f∈ W^r ∩Δ⁽²⁾, r∈ \mathbb N, there exists a number \mathcal N=\mathcal N(f,r), such that for every n≥ \mathcal N, there is an algebraic polynomial of degree ≤ n which is in Δ⁽²⁾ and such that ‖ (f-P_n)/(φ^r) ‖_∞ ≤ (c(r))/(n^r) ‖ f^(r)‖_∞ , where φ(x):= √(1-x²). For r=1 and r=2, the above result holds with \mathcal N=1 and is well known. For r≥ 3, it is not true, in general, with \mathcal N independent of f.

Related

• Interpolatory estimates for convex piecewise polynomial approximation
• Interpolatory pointwise estimates for monotone polynomial approximation
• Exact pointwise estimates for polynomial approximation with Hermite interpolation
• Onesided, intertwining, positive and copositive polynomial approximation with interpolatory constraints
• Uniform and pointwise shape preserving approximation (SPA) by algebraic polynomials: an update