Positive definite kernels on compact interval $[0,1]$

Question

From How to prove that a kernel is positive definite? I learned that a function $f:[0,\infty)\to\mathbb{R}$ induces a positive definite kernel $K:\mathbb{R}^2\to\mathbb{R}$, $K(x,y)=f((x-y)^2)$ if $f$ is completely monotonic. I want to know whether the same holds for a kernel with domain $[0,1]^2$.

So does a function $f:[0,1]\to\mathbb{R}$ induce a positive definite kernel $K:[0,1]^2\to\mathbb{R}$, $K(x,y)=f((x-y)^2)$ if $f$ is completely monotonic?

Answer 1

Unfortunately, the answer is no.

Consider $f(x)=1-x$ inducing $K(x,y)=1-(x-y)^2$. Then $(x_1,x_2,x_3)=(0,\frac12,1)$ gives $$(K(x_i,x_j))_{i,j=1}^{3}=\begin{pmatrix}1&\frac34&0\\\frac34&1&\frac34\\0&\frac34&1\end{pmatrix},$$ which has eigenvalue $1-\frac34\sqrt2<0$, showing $K$ is not positive definite.