Polynomial Approximation in $ L^2 $ of the Double Exponential via Complex Analysis

TL;DR

The paper studies the rate of polynomial approximation in $L^2(\mu_1)$ for the double-exponential weight $d\mu_1(x)=\tfrac{1}{2}e^{-|x|}\,dx$. It builds a complex-analytic framework based on Meixner-Pollaczek polynomials and their generating functions to relate coefficients $f_k$ to boundary values of holomorphic extensions, yielding sharp inequalities. The main result is a tight bound $\sum_{k\ge1} \log^2(e+k) f_k^2 \lesssim \int \log^2(e+|x|) f^2 \,d\mu_1 + \int (f')^2 \,d\mu_1$, implying a logarithmic rate of polynomial approximation, with a complementary bound and tensorization to $\mu_1^{\otimes d}$ discussed. The work highlights the power of complex-analysis methods in orthogonal polynomial approximation and proves optimality via explicit extremal constructions.

Abstract

We study the polynomial approximation problem in $L^2(μ_1)$ where $μ_1(dx) = e^{-|x|}/2 dx$. We show that for any absolutely continuous function $f$, $$ \sum_{k=1}^{\infty} \log^2(e+k) \langle f, P_k \rangle^2 \ \leq C \left( \int_{\mathbb{R}} \log^2(e+\lvert x \rvert) f^2 \, dμ_1 \ + \ \int_{\mathbb{R}} (f')^2 \, dμ_1 \right) $$ for some universal constant $C>0$, where $(P_k)_{k \in N}$ are the orthonormal polynomials associated with $μ_1$. This inequality is tight in the sense that $\log^2(e +k)$ on the left hand-side cannot be replaced by $a_k \log^2(e +k)$ with a sequence $a_k \longrightarrow \infty$. 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 $μ_1^{\otimes d}$ in $\mathbb{R}^d$ via a tensorization argument. Our proof relies on an explicit formula for the generating function of orthonormal polynomials associated with the weight $\frac{1}{2\cosh(πx/2)}$ and some complex analysis.

Figures (1)

• Figure 1: Illustration of the region $A(\varepsilon) = \arctan(D(0,1-\varepsilon))$ for $\varepsilon=0.1$. The dashed vertical lines correspond to $Re(z) = \pm \pi/4$ while the horizontal lines correspond to $Im(z) =\pm \frac{1}{2}\log R_{1-\epsilon} = \pm \frac{1}{2}\log R_{0,1-\epsilon}$ where $R_{\theta,r}$ is defined in Lemma \ref{['lem_467']} below.

Theorems & Definitions (31)

• Theorem 1
• Theorem 2
• proof
• Corollary 3
• Remark 4
• Lemma 5
• Lemma 6
• proof
• Lemma 7
• proof