Polynomial approximation on convex subsets of \mathbb Rⁿ

1999/10/28 by Y. Brudnyi, Yuri Brudnyi, Brudnyi, Y. +2
Mathematics · #41A10 #Advanced Banach Space Theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Fixed Point Theorems Analysis #Functional Analysis (math.FA) #Functional Equations Stability Results #math.CA #math.FA #msc:41A10

paper · pdf · doi:10.48550/arxiv.math/9910160

36 pages

arxiv created 1999/10/28 · openalex publication_date 1999/10/28 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be a closed bounded convex subset of \Bbb Rⁿ; then by a result of the first author, which extends a classical theorem of Whitney there is a constant w_m(K) so that for every continuous function f on K there is a polynomial ϕ of degree at most m-1 so that |f(x)-ϕ(x)|≤ w_m(K)sup_{x,x+mh∈ K} |Δ_h^m(f;x)|. The aim of this paper is to study the constant w_m(K) in terms of the dimension n and the geometry of K. For example we show that w₂(K)≤ \frac12[log₂n]+\frac54 and that for suitable K this bound is almost attained. We place special emphasis on the case when K is symmetric and so can be identified as the unit ball of finite-dimensional Banach space; then there are connections between the behavior of w_m(K) and the geometry (particularly the Rademacher type) of the underlying Banach space. It is shown for example that if K is an ellipsoid then w₂(K) is bounded, independent of dimension, and w₃(K)∼ log n. We also give estimates for w₂ and w₃ for the unit ball of the spaces ℓ_pⁿ where 1≤ p≤ ∞.

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