An analog of Rudin’s theorem for continuous radial positive definite functions of several variables

Let [inline-graphic not available: see fulltext] be the class of radial real-valued functions of m variables with support in the unit ball \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{B}$\end{document} of the space ℝm that are continuous on the whole space ℝm and have a nonnegative Fourier transform. For m ≥ 3, it is proved that a function f from the class [inline-graphic not available: see fulltext] can be presented as the sum \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sum {f_k \tilde *f_k } $\end{document} of at most countably many self-convolutions of real-valued functions fk with support in the ball of radius 1/2. This result generalizes the theorem proved by Rudin under the assumptions that the function f is infinitely differentiable and the functions fk are complex-valued.