Spectral analysis on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm SL}(2, \mathbb{R})}$$\end{document}

Let G be the group \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm SL}(2, \mathbb{R})}$$\end{document}. For this group we prove a version of Schwartz’s theorem on spectral analysis for the group G. We find the sharp range of Lebesgue spaces Lp(G) for which a smooth function is not mean periodic unless it is a cusp form. Failure of the Schwartz-like theorem is also proved when C∞(G) is replaced by Lp(G) with suitable p. We show that the last result is linked with the failure of the Wiener-tauberian theorem for G.