Sur la meilleure approximation de |x| par des polynomes de degrés donnés

By S. Bernstein. 1914

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Minimax Rates of Estimating Approximate Differential Privacy (2019, background): “p) 0 (dt) = r ce"plnn n L[[ t]+;[ 1;1]] & p^ s ce"p d2 2 nlnn : (95) The last inequality follows from the following lemma, with a choice of L= d 2 lnnfor some constant d 2. 33 Lemma 19 ([Ber14] ). For L>1, L[[ t]+;[ 1;1]] = L[jtj;[ 1;1]] = L 1(1 + o(1)) 1 L ; (96) where ˇ0:2802 is ”
Testing composite hypotheses, Hermite polynomials and optimal estimation of a nonsmooth functional (2011, methodology): “Then The coeﬃcients g ∗ 2 k and g 2 k satisfy for all 0 The uniform error bounds (26) and (27) were proved in Bernstein (1913).”
Multivariate polynomial approximation in the hypercube (2016, unknown): “The following result goes back to Bernstein’s prize-winning memoir of 1914 [2].”
Uniform approximation of sgn x by polynomials and entire functions (2006, background): “Bernstein [3, 4] who found that for the error Em(α) of the best approximation by polynomials of degree m the following limit exists:”