Compactly supported positive definite radial functions

We provide criteria for positive definiteness of radial functions with compact support. Based on these criteria we will produce a series of positive definite and compactly supported radial functions, which will be very useful in applications. The simplest ones arecut-off polynomials, which consist of a single polynomial piece on 0, 1] and vanish on 1, ∞). More precisely, for any given dimensionn and prescribedC ^k smoothness, there is a function inC ^k (? ⁿ ), which is a positive definite radial function with compact support and is a cut-off polynomial as a function of Euclidean distance. Another example is derived from odd-degreeB-splines.