Recurrence relations for radial positive definite functions

In the theory of radial basis functions as well as in the theory of spherically symmetric characteristic functions recurrence relations are used to construct d-dimensional functions starting with lower-dimensional ones. We show that the operators used so far are special cases of one step recurrence relations for 2-radial positive definite functions. We further give the analogue for 1-radial functions and thereby define a turning bands operator for 1-symmetric characteristic functions.  2002 Elsevier Science (USA). All rights reserved.