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

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