Best polynomial approximation on the unit ball

2016/09/18 by Miguel Pinar, Yuan Xu, Pinar, Miguel +1
Mathematics · #33C50 #42C10 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA #msc:33C50 #msc:42C10

paper · pdf · doi:10.48550/arxiv.1609.05515

Final form in IMA J. Numer. Anal

arxiv created 2017/05/01 · arxiv updated 2017/05/03

Abstract

Let E_n(f)_μ be the error of best approximation by polynomials of degree at most n in the space L²(\varpi_μ, \mathbbB^d), where \mathbbB^d is the unit ball in ℝ^d and \varpi_μ(x) = (1-‖x‖²)^μ for μ> -1. Our main result shows that, for s ∈ ℕ, E_n(f)_μ≤ c n^-2s[E_n-2s(Δ^s f)_μ+2s + E_n(Δ₀^s f)_μ], where Δ and Δ₀ are the Laplace and Laplace-Beltrami operators, respectively. We also derive a bound when the right hand side contains odd order derivatives.

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