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

Let $\mathscr G_m$ be the class of radial real-valued functions of $m$ variables with support in the unit ball $\mathbb B$ of the space $\mathbb R^m$ that are continuous on the whole space $\mathbb R^m$ and have a nonnegative Fourier transform. For $m\ge3$, it is proved that a function $f$ from the class $\mathscr G_m$ can be presented as the sum $\sum f_k\widetilde\ast f_k$ of self-convolutions of at most countably many real-valued functions $f_k$ 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 $f_k$ are complex-valued.