*2.2. Multiply Monotonicity*

Since Fourier transforms are not always easy to compute, it is convenient to decide whether a function is strictly positive definite and radial on R*n* by the multiply monotonicity for limited choices of *n*.

**Definition 3.** *A function ϕ* : (0, ∞) → R*, which is in <sup>C</sup>k*−<sup>2</sup>(0, <sup>∞</sup>), *k* ≥ 2*, and for which* (−<sup>1</sup>)*<sup>l</sup>ϕ*(*l*)(*r*) *is non-negative, non-increasing, and convex for l* = 0, 1, ···, *k* − 2*, is called k-times monotone on* (0, <sup>∞</sup>)*. In the case k* = 1*, we only require ϕ* ∈ *C*(0, ∞) *to be non-negative and non-increasing.*

This definition can be found in the monographs [2,3]. The following Micchelli theorem (see [4]) provides a multiply monotonicity characterization of strictly positive definite radial functions.

**Theorem 2.** *Let k* = *n*2 + 2 *be a positive integer. If ϕ* : [0, ∞) → R, *ϕ* ∈ *C*[0, <sup>∞</sup>)*, is k-times monotone on* (0, <sup>∞</sup>)*, but not constant, then ϕ is strictly positive definite and radial on* R*n for any n such that n*2 ≤ *k* − 2*.*
