The Hermite expansion of the characteristic functions

We calculate the coefficients of the Hermite expansion of the characteristic function of [-a,a]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[-a,a]$$\end{document} and [0, a], explicitly. As applications, we calculate the coefficients of the Hermite expansion of the sinc function, the Heaviside function (characteristic function of [0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,\infty )$$\end{document}) and derivatives of the Dirac’s delta functions.