# Reproducing Kernel Hilbert Space

Since writing about the simulation of fractional Brownian motion, I’ll spend an hour to write RKHS.

Definition: A Hilbert space $\mathcal{H}$ is a RKHS if $|\mathcal{F}_t[f]|=|f(t)|\leq M\|f\|_{\mathcal{H}}, \forall f\in \mathcal{H}$

Theorem: If $\mathcal{H}$ is a RKHS, then for each $t \in X$ there exists a function $K_t \in \mathcal{H}$ (called the representer of $t$) with the reproducing property

$\mathcal{F}_t[f]=_{\mathcal{H}}=f(t), \forall f\in\mathcal{H}$

Therefore, $K_t(t^{'})=_{\mathcal{H}}$.

Definition: $K: X\times X\rightarrow\mathbb{R}$ is a reproducing kernel if it’s symmetric and positive definite.

Theorem: A RKHS defines a corresponding reproducing kernel. Conversely, a reproducing kernel defines a unique RKHS.

Once we have the kernels, if $f(\cdot)=\sum \alpha_i K(t_i,\cdot), g(\cdot)=\sum \beta_i K(t^{'}_i,\cdot)$,

then $_{\mathcal{H}}=\sum\sum \alpha_i \beta_j K(t_i, t^{'}_j)$

When it comes to the fractional Brownian motion

Theorem: For fBm RKHS $K(x,x^{'})=R(x,x^{'})$ is symmetric and positive definite, , there exists $K^H(x,\cdot)$ s.t $K(x,x^{'})=\int K^H(x,y)K^H(x^{'},y)dy$

Take Wiener integral as an example:

$\begin{pmatrix} \textit{dual space}&\textit{inner product}&E&\leftrightarrow&<\cdot,\cdot>_{\mathcal{H}}&\leftrightarrow&<\cdot,\cdot>_{L^{2}}\\ \delta_{t}(\cdot)& &B^H_t&\leftrightarrow&R(\cdot,t)&\leftrightarrow&K^H(t,\cdot)\\ f(\cdot)& &\textit{"}\int_0^T f(t)B^H_tdt\textit{"}&\leftrightarrow&K^H\circ(K^H)^{*}f&\leftrightarrow&(K^H)^{*}f \end{pmatrix}$

where $K^H\circ f(t)=\int_0^T K^H(t,s)f(s)ds$,

$(K^H)^{*}\circ f(t)=\int_0^T K^H(s,t)f(s)ds$ and

$K^H\circ(K^H)^{*}f(t)=\int_0^T R(t,s)f(s)ds$

For example, when $H=\frac{1}{2}$, $K^H(t,s)=1_{[0,t]}(s),$, we can check

$E(B_tB_s)=_{\mathcal{H}}=<1_{[0,t]}(\cdot),1_{[0,s]}(\cdot)>_{L^2}$

Claim: By previous theorem, the reproducing kernel $R(t,s)$ uniquely defines a RKHS $L(R(t,\cdot))$ with the inner product $\mathcal{H}$