Open AccessAn L 2 form of Bernstein's inequality
Author(s) -
George E. Backus
Publication year - 1979
Publication title -
proceedings of the national academy of sciences of the united states of america
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 5.011
H-Index - 771
eISSN - 1091-6490
pISSN - 0027-8424
DOI - 10.1073/pnas.76.7.3061
Subject(s) - combinatorics , degree (music) , mathematics , polynomial , differentiable function , physics , unit sphere , function (biology) , mathematical analysis , evolutionary biology , acoustics , biology
SupposeP is ap th degree real polynomial function inn variables andf =P ǀS n -1is the restriction ofP to the unit sphereS n -1inRn . Bernstein's inequality asserts that ([unk]0 k f)2 +p 2 ([unk]0 k -1f )2 ≤p 2k ∥f ∥∞2 , wherek ≥ 1 and differentiation is with respect to arc length θ along any geodesic inS n -1. We find the constant corresponding top 2k when ∥f ∥∞ is replaced by ∥f ∥2 . One application is a condition on the coefficients of the expansion in surface spherical harmonics of anyg: S n -1→R , which condition suffices to assure thatg isk times differentiable.