# Kernels property: integral of kernel product $\propto k(x,y)$

Let $k$ be a kernel function (symmetric and semi-positive definite function).

Does the following relationship hold:

$\int_{-\infty}^{+\infty}k(x,u)k(y,u) du \propto k(x,y)$ ?

Or for what type of kernels does it hold?

I know that in the case of RBF Gaussian kernel it holds (see here: page 102 pdf; page 84 print; it relates to Gaussian processes; a positive answer would help me generalize this RBF property to any kernel function).