RKHS of discontinous function

Question

I have a discountinous function $f$ which I would like show it as $f \in \mathcal{H}_k(\mathcal{X})$ where $\mathcal{H}_k(\mathcal{X})$ is a RKHS generated by kernel $k$ in domain $\mathcal{X}$. Is it possible to define such a RKHS? If so what would be the candidate for the kernel?

Answer 1

Let $f:\mathcal{X}\to \mathbb{C}$. The kernel $K:\mathcal{X}\times \mathcal{X}\to \mathbb{C}$, given by $K(x,y)=\overline{f(x)}f(y)$ is positive definite, and $f\in \mathcal{H}_K$.

If you want a particular kernel, you need to give us more details on $\mathcal{X}$ and $f$.

Please see the book Integral Transforms, Reproducing Kernels and Their Applications, for instance.

You can find related results searching for "\(f(x)=\langle f,K^x\rangle\) reproducing kernel" on SearchOnMath.