Newtons law for gravity states that:


$$F\_{12} = \frac{G m\_1 m\_2} {|x\_1-x\_2|^2}$$


The function :


$$k(x,y):=\exp(-| x-y|^2)$$


is known to be a positive definite function, called [the RBF-kernel](https://en.wikipedia.org/wiki/Radial_basis_function_kernel).


It follows that


$$k(x\_1,x\_2) := \exp( -|x\_1-x\_2|^2) = \exp(-\frac{G m\_1 m\_2}{F\_{12}})$$


is a positive definite , symmetric function. My question, is if you know of any other situations, where terms like


$$\exp(-\frac{m\_1 m\_2 G}{F\_{12}})$$


are shown to be positive definite kernels and what is the explanation for this?


(By this I mean: Do you know of situations where one proves that functions like:


$$f(a,b) = \exp(-\frac{m\_a m\_b G}{F\_{ab}})$$


are positive definite - and how is it proved? Here $m\_x$ denotes numbers $>0$ and $F\_{xy}$ is a function of $x,y$.)


Thanks for your help.



Inverse square-law as a positive definite kernel?
Please look pages 141-147 of the book:


S. Saitoh and Y. Sawano, Theory of Reproducing Kernels and Applications,
Developments in Mathematics 44, Springer (2016).
2023.5.4.20:36　


