Exact pointwise estimates for polynomial approximation with Hermite interpolation

2020/06/04 by Kirill A. Kopotun, Kopotun, Kirill A., D. Leviatan +3
Mathematics · #41A10 #41A17 #41A28 #Approximation Theory and Sequence Spaces #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical functions and polynomials #Primary 41A25 #Secondary 41A05

paper · pdf · doi:10.48550/arxiv.2006.03126

openalex publication_date 2020/06/04 · openalex created_date 2020/06/12 · openalex updated_date 2026/07/28

Abstract

We establish best possible pointwise (up to a constant multiple) estimates for approximation, on a finite interval, by polynomials that satisfy finitely many (Hermite) interpolation conditions, and show that these estimates cannot be improved. In particular, we show that \bf any algebraic polynomial of degree n approximating a function f∈ C^r(I), I=[-1,1], at the classical pointwise rate ρ_n^r(x) ω_k(f^(r), ρ_n(x)), where ρ_n(x)=n^-1√(1-x²)+n^-2, and (Hermite) interpolating f and its derivatives up to the order r at a point x₀∈ I, has the best possible pointwise rate of (simultaneous) approximation of f near x₀. Several applications are given.

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