Introduction:
Let $\mathcal{H}$ be a Hilbert space of functions $\Omega\to\mathbb{R}$ with reproducing Kernel $K:\Omega\times\Omega\to\mathbb{R},\,\Omega\subset\mathbb{R}^d,\, d>1$, where $K$ is continuous and $\Omega$ is a bounded domain. You can imagine $\mathcal{H}$ as the closure of the span of translates $K(\cdot,x),\, x\in\Omega$ with respect to the inner product $$(f,g)_K:=\sum_{j=1}^n\sum_{k=1}^m\alpha_j\beta_k K(x_j,x_k),$$ where $$f=\sum_{j=1}^n\alpha_jK(\cdot,x_j),\;g=\sum_{k=1}^m\beta_kK(\cdot,x_k)\in \text{span}\{K(\cdot,x)\mid x\in\Omega\},$$ i.e $$\mathcal{H}\equiv\mathcal{H}(\Omega,K):=\overline{\text{span}\{K(\cdot,x)\mid x\in\Omega\}}^{\|\cdot\|_K}.$$
Now we want to reconstruct functions $f\in\mathcal{H}$ on a finite set of distinct points $X_n:=\{x_1,\ldots,x_n\}\subset\Omega$ by data dependent Basis functions that are given by translates of the Kernel, i.e. $K(\cdot,x_j),\, 1\le j\le n$. The interpolant should then be represented by a linear combination of those. That implies the data dependent subspace $$\mathcal{H}_{X_n}:=\text{span}\{K(\cdot,x_j)\mid 1\le j\le n\}\subset\mathcal{H}$$ in which we are looking for the interpolant $s^*_n\in\mathcal{H}_{X_n}$ for some $f\in\mathcal{H}$ on the point set $X_n$ such that $s^*_n(x_j)=f(x_j)$ for every $1\le j\le n$.
Now the unique interpolant $s^*_n\in\mathcal{H}_{X_n}$ is given by the orthogonal projection of $f\in\mathcal{H}$ onto the finite dimensional subspace $\mathcal{H}_{X_n}$. We define the corresponding Operator by $$\Pi_n:\mathcal{H}\to\mathcal{H}_{X_n},\, f\mapsto s^*_n$$ and it would be desirable to prove some convergence $\|\Pi_n(f)-f\|_K\to 0$ for $n\to\infty$. To do so, we consider a sequence of finite point sets $$\{x_1\}:=X_1\subset X_2\subset\ldots\subset X_n\subset X_{n+1}\subset\ldots\subset\Omega$$ and we have to assume that the filling density $h(X_n,\Omega)$ goes to $0$ for $n\to\infty$, i.e. $$h(X_n,\Omega):=\sup_{\omega\in\Omega}\min_{x\in X_n}\|\omega-x\|_2\xrightarrow{n\to\infty}0,$$ where $\|\cdot\|_2$ denotes the standard norm on $\mathbb{R}^d$. If we assume this we get $$\overline{X_\infty}^{\|\cdot\|_2}:=\overline{\bigcup_{n\in\mathbb{N}}X_n}^{\|\cdot\|_2}=\Omega.$$ Now for $y\in\Omega$ we get a sequence $(x_n)_{n\in\mathbb{N}}\subset\Omega$ such that $x_n\in X_n$ for every $n\in\mathbb{N}$ and $x_n\to y$ for $n\to\infty$ and it follows that
\begin{align*} \|K(\cdot,x_n)-K(\cdot,y)\|_K^2&=(K(\cdot,x_n),K(\cdot,x_n))_K-2(K(\cdot,x_n),K(\cdot,y))_K\\ &\quad+(K(\cdot,y),K(\cdot,y))_K\\ &=K(x_n,x_n)-2K(x_n,y)+K(y,y)\\ &\xrightarrow{n\to\infty}2K(y,y)-2K(y,y)=0. \end{align*}
We can also select an arbitrary subset $Y:=\{y_1,\ldots,y_N\}\subset\Omega$ und sequences $(x_n^{(j)})_{n\in\mathbb{N}}$ such that $x_{n}^{(j)}\to y_j,\,1\le j\le N$ and functions
$$g=\sum_{j=1}^N c_j K(\cdot,y_j)\in\mathcal{H}_Y\quad\text{and}\quad t=\sum_{j=1}^N c_j K(\cdot,x_{n}^{(j)})\in\mathcal{H}_{X_n},$$ where $c=(c_1,\ldots,c_N)^T\in\mathbb{R}^N$. Now we finally get
\begin{align*}
\|\Pi_n(g)-g\|_K&=\inf_{s\in\mathcal{H}_{X_n}}\|s-f\|\le\|t-g\|_K =\left\|\sum_{j=1}^N c_j\left(K(\cdot,x_n^{(j)})-K(\cdot,y_j)\right) \right\|_K\\
&\le \sum_{j=1}^N|c_j|\cdot\underbrace{\|K(\cdot,x_n^{(j)})-K(\cdot,y_j)\|_K}_{\to 0}\xrightarrow{n\to\infty} 0.
\end{align*}
We have now shown the claimed convergence for the dense subset
$$\bigcup_{Y\subset\Omega}\mathcal{H}_{Y}=\text{span}\{K(\cdot,x)\mid x\in\Omega\}\subset\mathcal{H}$$
and it can now be continuously be extended to $\mathcal{H}$.
Questions:
Since the orthogonal projection in general is given by $$\sum_{j=1}^n \frac{(s_j,f)}{\|s_j\|^2}s_j,$$ if $\{s_1,\ldots,s_n\}$ is an orthogonal basis, I want to proof the following statement:
If $S=\{s_j\mid j\in\mathbb{N}\}\subset\mathcal{H}$ is an orthogonal complete system in $\mathcal{H}$, i.e. $(s_j,s_k)_K=0,\, j\neq k$ and its span is dense in $\mathcal{H}$, we get a series expansion $$f=\sum_{j=1}^\infty \frac{(s_j,f)_K}{\|s_j\|_K}s_j\quad\text{for every }f\in\mathcal{H}.$$ (wich is also pointwise true since convergence in $\|\cdot\|_K$ implies pointwise convergence, but that's not important here.)
Well... And here it's getting down hill... In the first Theorem, I prove the convergence $\|\Pi_n(f)-f\|_K\to 0$ but I never mentioned any complete systems nor orthogonality. I basically worked with the standard basis $T_n:=\{K(\cdot,x_j)\mid 1\le j\le n\}$ of the spaces $\mathcal{H}_{X_n}$ and obviously $\{K(\cdot,x)\mid \Omega\}$ is a complete system in $\mathcal{H}$ since we basically identified $\mathcal{H}$ by the closure of the span of this set. The problem now is the change of basis.
Right now in this state I can't directly follow this series expansion from the convergence I just proved. I need another theorem in between... Something like:
If $S=\{s_j\mid j\in\mathbb{N}\}$ is complete in $\mathcal{H}$ then it "arises" from a sequence $X_1\subset X_2\subset\ldots\subset\Omega$ with $h(X_n,\Omega)\to 0$ and corresponding spaces $\mathcal{H}_{X_1}\subset\mathcal{H}_{X_2}\subset\ldots\subset\mathcal{H}$ such that $S_n:=\{s_1,\ldots,s_n\}\subset S$ is a Basis for $\mathcal{H}_{X_n}$ for every $n\in\mathbb{N}$.
Then $S$ would be a basis for $\bigcup_{n\in\mathbb{N}}\mathcal{H}_{X_n}$ and I think that one would be dense in $\mathcal{H}$. Not sure about that one but since $\bigcup_{n\in\mathbb{N}}X_n$ is dense in $\Omega$ I would guess $$\overline{\bigcup_{n\in\mathbb{N}}\mathcal{H}_{X_n}}=\overline{\text{span}\{K(\cdot,x)\mid x\in\bigcup_{n\in\mathbb{N}}X_n\}}=\text{span}\{K(\cdot,x)\mid x\in\Omega\}$$ and $$\overline{\text{span}\{K(\cdot,x)\mid x\in\Omega\}}=\mathcal{H}$$ where the second one is just the definition of $\mathcal{H}$ basically. The Problem here is, that I am not sure about the following.
When we assume the space $\mathcal{H}_{X_n}=\text{span}\{K(\cdot,x_j)\mid x_j\in X_n,\,1\le j\le n\}$ and we increase the dimension by adding one point so the set $X_n$, so we get the set $X_{n+1}=X_n\cup\{x_{n+1}\}$, we get the space $\mathcal{H}_{X_{n+1}}$. But now lets consider an alternative basis $S_n=\{s_1,\ldots,s_n\}$ of $\mathcal{H}_{X_n}$ and increase the point set by one point. We can now construct an alternative basis $B_{n+1}=\{b_1,\ldots,b_{n+1}\}$ of $\mathcal{H}_{X_{n+1}}$. Is $s_j=b_j$ for every $1\le j\le n$, i.e. $S_n\subset B_{n+1}$??? I don't think it is to be honest...
Is there maybe another way to get from that convergence to the claimed statement about orthogonal series expansions? Maybe I am thinking to complicate... Thank you very much for your help! Kind regards.