Uniform Approximation by Polynomials with Integer Coefficients via the Bernstein Lattice

2023/11/17 by C. Si̇nan Güntürk, Güntürk, C. Sinan, Weilin Li +1
Computer Science · Mathematics · #41A29 #Advanced Topology and Set Theory #Approximation Theory and Sequence Spaces #Classical Analysis and ODEs (math.CA) #Digital Filter Design and Implementation #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2311.10901

openalex publication_date 2023/11/17 · openalex created_date 2023/11/22 · openalex updated_date 2026/07/28

Abstract

Let \mathscrC_ℤ([0,1]) be the metric space of real-valued continuous functions on [0,1] with integer values at 0 and 1, equipped with the uniform (supremum) metric d_∞. It is a classical theorem in approximation theory that the ring ℤ[X] of polynomials with integer coefficients, when considered as a set of functions on [0,1], is dense in \mathscrC_ℤ([0,1]). In this paper, we offer a strengthening of this result by identifying a substantially small subset \bigcup_n \mathscrB_n of ℤ[X] which is still dense in \mathscrC_ℤ([0,1]). Here \mathscrB_n, which we call the ``Bernstein lattice,'' is the lattice generated by the polynomials p_n,k(x) := \binomnk x^k(1-x)^n-k, ~~k=0,…,n. Quantitatively, we show that for any f ∈ \mathscrC_ℤ([0,1]), d_∞(f, \mathscrB_n) ≤ (9)/(4) ω_f(n^-1/3) + 2 n^-1/3, ~~n ≥ 1, where ω_f stands for the modulus of continuity of f. We also offer a more general bound which can be optimized to yield better decay of approximation error for specific classes of continuous functions.

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