Optimal-Degree Polynomial Approximations for Exponentials and Gaussian Kernel Density Estimation

@article{Aggarwal2022OptimalDegreePA,
  title={Optimal-Degree Polynomial Approximations for Exponentials and Gaussian Kernel Density Estimation},
  author={Amol Aggarwal and Josh Alman},
  journal={ArXiv},
  year={2022},
  volume={abs/2205.06249}
}
For any real numbers B ≥ 1 and δ ∈ (0 , 1) and function f : [0 , B ] → R , let d B ; δ ( f ) ∈ Z > 0 denote the minimum degree of a polynomial p ( x ) satisfying (cid:12)(cid:12) p ( x ) − f ( x ) (cid:12)(cid:12) < δ for each x ∈ [0 , B ]. In this paper, we provide precise asymptotics for d B ; δ ( e − x ) and d B ; δ ( e x ) in terms of both B and δ , improving both the previously known upper bounds and lower bounds. In particular, we show that and we explicitly determine the leading coe… 

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