A question on the product of positive definite functions Show that if $f$ and $g$ are continuous positive definite functions in $\mathbf{R}^1$, then $f(x)g(y)$ is positive definite on $\mathbf{R}^2$. 
I just wanted to check if the approach I'm using is correct, 
I defined the matrix $A_{j,k}=f(x_j-x_k)$ and $B_{j,k}=g(y_j-y_k)$ are positive definite functions for all $x_1 \cdots x_N$ and $y_1 \cdots y_N$. Would this imply that $A_{j,k} B_{j,k}=f(x_j-x_k)g(y_j-y_k)$ is also positive definite? 
Also, how exactly do I use Bochner's Theorem to show the following? 
Really appreciate the help, thanks!

Bochner theorem states that $f$ is positive definite iff $f = \widehat{\mu}$, with $\mu$ a positive Radon measure. If $f_1$, $f_1$ are positive definite then:
$$
  f_1 \cdot f_2 = \widehat{\mu_1} \cdot \widehat{\mu_2} = \big(\mu_1 \ast \mu_2 \big)^\wedge,
$$
and the convolution of two positive measures is positive.
