Radial basis function

Remarks

Common choices of $\varphi$, with a shape parameter $\varepsilon > 0$:

1. Gaussian: $\varphi(r) = e^{-(\varepsilon r)^2}$;
2. multiquadric: $\varphi(r) = \sqrt{1 + (\varepsilon r)^2}$;
3. inverse multiquadric: $\varphi(r) = 1 / \sqrt{1 + (\varepsilon r)^2}$;
4. polyharmonic splines: $\varphi(r) = r^k$ for odd $k$ and $\varphi(r) = r^k \log r$ for even $k$, with $\varphi(0) = 0$.

Example: $\Phi(x) = e^{-\norm{x}^2}$ on $\R^2$ is radial: $\Phi(x_1, x_2) = e^{-(x_1^2 + x_2^2)}$ only depends on $\norm{x}$. For instance $\Phi(1, 0) = \Phi(0, 1) = e^{-1}$.

Non-example: $\Psi(x_1, x_2) = e^{-(x_1^2 + 2 x_2^2)}$ is not radial: $\norm{(1, 0)} = \norm{(0, 1)} = 1$, but $\Psi(1, 0) = e^{-1} \ne e^{-2} = \Psi(0, 1)$.