2023/10/15 by Sean Richardson, Richardson, Sean · 1 citation
Engineering · Mathematics · #42A05 (Primary) 26C05 #65D10 (Secondary) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Numerical methods in inverse problems #Reservoir Engineering and Simulation Methods #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2310.09713
openalex publication_date 2023/10/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We consider functions f: ℤ → ℝ and kernels u: \-n, ⋯, n\ → ℝ normalized by ∑ℓ = -nn u(ℓ) = 1, making the convolution u ∗ f a "smoother" local average of f. We identify which choice of u most effectively smooths the second derivative in the following sense. For each u, basic Fourier analysis implies there is a constant C(u) so ‖Δ(u ∗ f)‖ℓ2(ℤ) ≤ C(u)‖f‖ℓ2(ℤ) for all f: ℤ → ℝ. By compactness, there is some u that minimizes C(u) and in this paper, we find explicit expressions for both this minimal C(u) and the minimizing kernel u for every n. The minimizing kernel is remarkably close to the Epanechnikov kernel in Statistics. This solves a problem of Kravitz-Steinerberger and an extremal problem for polynomials is solved as a byproduct.