The Fourier Formula for Discontinuous Functions of Several Variables

An analog of the classical Fourier formula for the characteristic function χΩ of a convex compact set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\Omega \subset {\mathbb{R}}^m $$ \end{document} is considered: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\chi _\Omega (x_0 ) = {\mathop {\lim }\limits_{R \to {\text{ + }}\infty }} \int\limits_{RW} {\hat \chi _\Omega (y)e^{2\pi ix_0 \cdot y} dy,} $$ \end{document} where W is a polyhedral in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${\mathbb{R}}^m $$ \end{document}. Bibliography: 6 titles.