I am going through the fourth volume of Gelfand and Vilenkin's text Generalized Functions. They introduce "conditionally positive definite generalized functions" which I've been unable to find much about in other books or online despite the authors stressing their importance. Their definition is given as follows:
We will call a generalized function $F$ a conditionally positive-definite generalized function of order $s$ if the inequality $(D\overline{D}F, \varphi * \varphi^*) \geq 0$ holds for all test functions $\varphi(x)$ and all linear homogeneous constant-coefficient differential operators $D$ of order $s$.
Here $D = \sum_{|k| = s} a_k (d^k/dx^k)$, $\overline{D} = (-1)^s\sum_{|k| = s} \overline{a}_k (d^k/dx^k)$, and $\varphi*\varphi^*$ is the convolution of $\varphi$ with $\varphi^*(x) = \overline{\varphi(-x)}$.
They also give another definition but the two are equivalent by equating differential operators with multiplication operators in the usual way using the Fourier transform so I've omitted it.
The notion of a positive-definite distribution is still commonly used today but I have not seen the prefix "conditionally" used anywhere. Is there a modern name for what they describe above? Or has it fallen in popularity since the book was written? If the latter is true I find this interesting since the authors stress these objects are very important for random processes.