Polynomial Approximation in L² of the Double Exponential via Complex Analysis

2025/02/11 by Pierre Bizeul, Bizeul, Pierre, Bo’az Klartag +1 · 1 citation
Mathematics · #Approximation Theory and Sequence Spaces #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Approximation and Integration #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.2502.07448

openalex publication_date 2025/02/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

Abstract

We study the polynomial approximation problem in L²(μ₁) where μ₁(dx) = e^-|x|/2 dx. We show that for any absolutely continuous function f, ∑_k=1^∞ log²(e+k) ⟨ f, P_k ⟩² ≤ C ( ∫_ℝ log²(e+| x |) f² dμ₁ + ∫_ℝ (f')² dμ₁ ) for some universal constant C>0, where (P_k)_{k ∈ N} are the orthonormal polynomials associated with μ₁. This inequality is tight in the sense that log²(e +k) on the left hand-side cannot be replaced by a_k log²(e +k) with a sequence a_k \longrightarrow ∞. When the right hand-side is bounded this inequality implies a logarithmic rate of approximation for f, which was previously obtained by Lubinsky. We also obtain some rates of approximation for the product measure μ₁^{⊗ d} in ℝ^d via a tensorization argument. Our proof relies on an explicit formula for the generating function of orthonormal polynomials associated with the weight (1)/(2\cosh(πx/2)) and some complex analysis.

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