# How to prove that Radial Basis Function can be derived by mapping function?

How to prove the radial basis function $k(u,v) = \int_{\mathbb{R}^d} \phi_t(u)\phi_t(v)dt$ can be integrated out by mapping function? $$\phi_{t}(u) = \frac{1}{(2\pi\Sigma)^{d/2}} \exp\left\{-\frac{\|u - t\|^2}{2\Sigma}\right\}$$ In other words, the integration will produce the radial basis function $$k(u,v) = \alpha \exp\left\{-\frac{\|u - v\|^2}{\beta}\right\}$$

Hope the problem is clear, it's my first time posting a question here.

• (+1) Because this is a homework/self-study question, you should add the self-study tag to your question. – Patrick Coulombe Feb 27 '15 at 5:23
• – Carl Nov 3 '17 at 13:55