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

Abstract

We study the polynomial approximation problem in $L^2(\hat{I}{¼}_1)$ where $\hat{I}{¼}_1(dx)=e^{-|x|}/2dx$. We show that for any absolutely continuous function $f$, $$\sum _{k=1}^{\infty }{\log }^2(e+k)⟨f,P_k{⟩}^2\le C\left({\int }_{\mathbb{R}}{\log }^2(e+|x|)f^{2}d\hat{I}{¼}_1+{\int }_{\mathbb{R}}(f^{\prime }{)}^{2}d\hat{I}{¼}_1\right)$$ for some universal constant $C>0$, where $(P_k{)}_{k\in N}$ are the orthonormal polynomials associated with $\hat{I}{¼}_1$. 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 $a_k⟶\infty$. 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 $\hat{I}{¼}_1^{\otimes d}$ 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 $\frac{1}{2\cosh (\ddot{I}€x/2)}$ and some complex analysis.

34 pages, 1 figure