Is $2^{xy}$ a positive definite kernel?

Question

Is $2^{xy}$ a positive definite kernel on $\mathbb{N}$?

i.e. for all $a_1, ..., a_n \in \mathbb{R}$, for all $x_1, ..., x_n \in \mathbb{N}$, $\sum_{i,j} a_i a_j 2^{x_ix_j}\geqslant 0$

Answer 1

If $K$ is a positive definite kernel, then $exp(K)$ is a positive definite kernel too (because it is the limit of a sum of positive definite kernel).

Because $2^{xy} = e^{xyln(2)}$, $2^{xy}$ is a positive definite kernel.