  Characteristic functions that are radially symmetric have a dual
interpretation, as they can be used as the isotropic correlation functions of
spatial random fields. Extensions of isotropic correlation functions from balls
into $d$-dimensional Euclidean spaces, $\R^{d}$, have been understood after
Rudin. Yet, extension theorems on product spaces are elusive, and a
counterexample provided by Rudin on rectangles suggest that the problem is
quite challenging. This paper provides extension theorem for multiradial
characteristic functions that are defined in balls embedded in $\R^d$ cross,
either $\R^{\dd}$ or the unit sphere $\S^{\dd}$ embedded in $\R^{\dd+1}$, for
any two positive integers $d$ and $\dd$. We then examine Turning Bands
operators that provide bijections between the class of multiradial correlation
functions in given product spaces, and multiradial correlations in product
spaces having different dimensions. The combination of extension theorems with
Turning Bands provides a connection with random fields that are defined in
balls cross linear or circular time.
