An analog of Rudin's theorem for continuous radial positive definite functions of several variables (Q483407)

A function \(f:{\mathbb{R}}^m\to {\mathbb{R}}\) is called radial if the value \(f(x)\) depends only on the norm of \(x\). Let \({\mathcal{G}}_m\) be the class of continuous radial functions with support in the unit ball and nonegative Fourier transform. The main result of the paper says that any \(\varphi \in {\mathcal{G}}_m\) can be represented as the sum \(\sum f_k\ast f_k\) of self convolutions of real-valued functions \(f_k\in L_2(\mathbb{R}^m)\) with supports in the ball of radius 1/2. This is a generalization of the result of \textit{W. Rudin} [Duke Math. J. 37, 49--53 (1970; Zbl 0194.36001)] from 1970.