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Characteristic functions that are radially symmetric have a dualinterpretation, as they can be used as the isotropic correlation functions ofspatial random fields. Extensions of isotropic correlation functions from ballsinto $d$-dimensional Euclidean spaces, $\R^{d}$, have been understood afterRudin. Yet, extension theorems on product spaces are elusive, and acounterexample provided by Rudin on rectangles suggest that the problem isquite challenging. This paper provides extension theorem for multiradialcharacteristic functions that are defined in balls embedded in $\R^d$ cross,either $\R^{\dd}$ or the unit sphere $\S^{\dd}$ embedded in $\R^{\dd+1}$, forany two positive integers $d$ and $\dd$. We then examine Turning Bandsoperators that provide bijections between the class of multiradial correlationfunctions in given product spaces, and multiradial correlations in productspaces having different dimensions. The combination of extension theorems withTurning Bands provides a connection with random fields that are defined inballs cross linear or circular time.

Rudin Extension Theorems on Product Spaces, Turning Bands, and Random Fields on Balls cross Time