Let G be the class of radial real-valued functions of m variables with support in the unit ball of the space that are continuous on the whole space and have a nonnegative Fourier transform. For , it is proved that a function f from the class G can be presented as the sum 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.