Smoothing of weights in the Bernstein approximation problem

2016/11/21 by Bakan, Andrew, Prestin, Jürgen
#32A15 #32A60 #41A10 #46E30 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1611.06708

Abstract

In 1924 S.Bernstein asked for conditions on a uniformly bounded on ℝ Borel function (weight) w: ℝ → [0, +∞ ) which imply the denseness of algebraic polynomials P in the seminormed space C⁰_w defined as the linear set \f ∈ C (ℝ) | w (x) f (x) → 0 as |x| → +∞\ equipped with the seminorm ‖f‖_w := sup_x ∈ ℝ w(x)| f( x )|. In 1998 A.Borichev and M.Sodin completely solved this problem for all those weights w for which P is dense in C⁰_w but there exists a positive integer n=n(w) such that P is not dense in C⁰_{(1+x²)ⁿ w}. In the present paper we establish that if P is dense in C⁰_{(1+x²)ⁿ w} for all n ≥ 0 then for arbitrary ε > 0 there exists a weight W_ε ∈ C^∞ (ℝ) such that P is dense in C ⁰_(1+x²)ⁿ W_ε for every n ≥ 0 and W_ε (x) ≥ w (x) + e^{- ε |x|} for all x∈ ℝ.

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• Bernstein's problem on weighted polynomial approximation
• De Branges' theorem on approximation problems of Bernstein type
• On weighted polynomial approximation
• Polynomials with exponents in compact convex sets and associated weighted extremal functions -- The Bernstein-Walsh-Siciak theorem
• Polynomial approximation in L_p(R, dμ). I