Is the matrix of exponential kernel with $L^1$ norm positive definite?

Question

Given an exponential function $f(x) = \exp[-(q_1|x_1|+\dots+q_d|x_d|)]$ for $x\in \mathbf{R}^d$ with $q_i > 0 \;(i=1,\dots,d)$.

For any distinct points $y_1,\dots,y_n\in\mathbf{R}^d$, define the $(i,j)$-entry of matrix $A$ by $A_{i,j}=f(y_i-y_j)$, is $A$ positive definite ? If so, how to prove this ?

Answer 1

$f$ is the Fourier transform of a positive integrable function. This implies that $f$ is positive definite (i.e., the matrix $(A_{i,j})$ is definite positive for all choices of $y_i$.) This is the easy part of Bochner's theorem.