of ${\mathbb{H}}$ satisfying: $$\sum_{m=0}^{\text{dim}{({\mathbb{H}})}}{\lvert\Phi_m(x)\rvert}^2<\infty$$ for all $x\in\Omega$. Let ${\mathfrak{H}}$ be another Hilbert space such that $\text{dim}({\mathbb{H}})=\text{dim}({\mathfrak{H}})$. Let $\{\phi_m\}_{m=0}^{\text{dim}{({\mathfrak{H}})}}$ be an orthonormal basis of ${\mathfrak{H}}$. Define $$\label{rep}
K(x,y)=\sum_{m=0}^{\text{dim}{({\mathbb{H}})}}\overline{\Phi_m(x)}\Phi_{m}(y).$$ Then $K(x,y)$ is a reproducing kernel, that is, $K(x,y)$ satisfies

1.  hermiticity, $K(x,y)=\overline{K(y,x)}$ for all $x,y\in \Omega$;

2.  positivity, $K(x,x)\geq 0$ for all $x\in\Omega$;

3.  idempotence, $\int_{\Omega}K(x,y)K(y,z)d\mu(y)=K(x,z),$

and ${\mathbb{H}}$ is the corresponding reproducing kernel Hilbert space. For $x\in \Omega$, define $$\label{rep-cs}
\mid x\rangle=K(x,x)^{-\frac{1}{2}}\sum_{m=0}^{\text{dim}{({\mathbb{H}})}}\overline{\Phi_m(x)}\phi_m \in \mathfrak H\; .$$ Then, $$\langle x\mid x\rangle=K(x,x)^{-1}
\sum_{m=0}^{\text{dim}{({\mathbb{H}})}}\overline{\Phi_m(x)}\Phi_m(x)=1,$$ and $$\mathcal{W}:{\mathfrak{H}}\longrightarrow{\mathbb{H}}\;\;\;\text{with}\;\;\;\mathcal{W}\phi(x)
=K(x,x)^{\frac{1}{2}}\langle x\mid \phi\rangle$$ is an isometry. Then, for $\phi,\psi\in{\mathfrak{H}}$ we have $$\begin{aligned}
\langle\phi\mid\psi\rangle_{{\mathfrak{H}}}=\langle\mathcal{W}\phi\mid\mathcal{W}\psi\rangle_{{\mathbb{H}}}&=&
\int_{\Omega}\overline{\mathcal{W}\phi(x)}\mathcal{W}\psi(x)d\mu(x)\\
&=&\int_{\Omega}\langle\phi\mid x\rangle\langle x\mid\psi\rangle K(x,x)d\mu(x),\end{aligned}$$ and $$\label{res-rep}
\int_{\Omega}\mid x\rangle\langle x\mid K(x,x)d\mu(x)=I_{{\mathfrak{H}}},$$ where $K(x,x)$ plays the role of a positive weight function. Thus, the set of states $\{\mid x\rangle \mid x\in\Omega\}$ forms a set of (generalized) CS.

In the case where $\{\Phi_m\}_{m=0}^{\text{dim}{({\mathbb{H}})}}$ is an orthogonal basis of ${\mathbb{H}}$, one can define $\rho(m)=\|\Phi_m\|^2;\;m=0,1,2, ...,\text{dim}{({\mathbb{H}})},$ and obtain an orthonormal basis $$\left\{\frac{\Phi_m}{\sqrt{\rho(m)}}\right\}_{m=0}^{\text{dim}{({\mathbb{H}})}}$$ of ${\mathbb{H}}$. Then, setting $$\label{rep-cs-2}
\mid x\rangle=K(x,x)^{-\frac{1}{2}}
\sum_{m=0}^{\text{dim}({\mathfrak{H}})}\frac{\Phi_m(x)}{\sqrt{\rho(m)}}\phi_m\in{\mathfrak{H}},$$ one obtains the desired result which is analogous to (\[can-cs\]).

The above discussion motivates the following definition.

\[def-CS\] Let ${\mathfrak{D}}$ be an open subset of $\mathbb C$. Let $$\Phi_m: {\mathfrak{D}}\longrightarrow {\mathbb{C}},\;\;\;m=0,1,2,\dots,$$ be a sequence of complex functions. Define $$\label{ma-cs}
\mid 