Approximation with one-bit polynomials in Bernstein form
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We prove various approximation theorems with polynomials whose coefficients with respect to the Bernstein basis of a given order are all integers. In the extreme, we draw the coefficients from the set {± 1} only. We show that for any Lipschitz function f:[0,1] → [-1,1] and for any positive integer n, there are signs σ_0,…,σ_n ∈{± 1} such that |f(x) - ∑_k=0^n σ_k nk x^k (1-x)^n-k | ≤C (1+|f|_Lip)/1+√(nx(1-x)) x ∈ [0,1]. These polynomial approximations are not constrained by saturation of Bernstein polynomials, and we show that higher accuracy is indeed achievable for smooth functions: If f has a Lipschitz (s-1)st derivative, then accuracy of order O(n^-s/2) is achievable with ± 1 coefficients provided f _∞ < 1, and accuracy of order O(n^-s) is achievable with unrestricted integer coefficients. Our approximations are constructive in nature.
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