**Definitions and setting**


Let $\mathcal{H}$ be a separable, infinite-dimensional, reproducing kernel Hilbert space on a nonempty set $X$. As usual, denote the reproducing kernel on $\mathcal{H}$ by $K$ and let $K\_x:=K(x,\cdot)$.


Let $M(\mathcal{H})$ denote the **multiplier algebra** of $\mathcal{H}$, that is, the functions $f:H\rightarrow \mathbb{R}$ satisfying: for all $h\in \mathcal{H}$ one has $x\mapsto f(x)h(x)$ is a function belonging to $H$ (i.e. pointwise multiplication).


A sequence $(k\_i)\_{i\in \mathbb{N}}\subseteq X^{\mathbb{N}}$ is said to be an **interpolating sequence** of $M(\mathcal{H})$ if, for any bounded sequence of reals $(r\_i)\_{i\in \mathbb{N}} \in \ell^{\infty}$ one can always find a multiplier $\mu\in M(\mathcal{H})$ satisfying:
$$
\phi(k\_i) = w\_i \mbox{ for all i in }\mathbb{N}
.
$$
As discussed in the recent article [Tsikalas - Interpolating sequences for pairs of spaces](https://doi.org/10.1016/j.jfa.2023.110059), many RKHSs have the property that the set of interpolating sequences can be characterized as the set of sequences $\boldsymbol{k}:=(k\_i)\in X^{\mathbb{N}}$ for which $T\_{\boldsymbol{k}}(\mathcal{H})=\ell^2$ where
$$
\begin{aligned}
T\_{\boldsymbol{k}}:\mathcal{H} & \rightarrow \mathbb{R}^{\mathbb{N}}\\
T\_{\boldsymbol{k}}(f) & := \big(\frac{f(k\_i)}{\|K\_{k\_i}\|}\big)\_{i\in \mathbb{N}}
\end{aligned}
$$
for all $f\in \mathcal{H}$.




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I recently came across this very old [MO post](https://mathoverflow.net/questions/126288/do-kernels-provide-a-basis-for-a-rkhs "Do kernels provide a basis for a RKHS?"), claiming that, $(K\_{k\_i})\_{i\in \mathbb{N}}$ is a Schauder, or even a Riesz, basis for $\mathcal{H}$ if $(k\_i)\_{i\in \mathbb{N}}$ is an interpolating sequence of $M(\mathcal{H})$. If this is correct, why is this this the case?




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**Intuition:**
I expect this somehow follows from the characterization of "good interpolating sequences", by which I mean, those satisfying $T\_{\boldsymbol{k}}(\mathcal{H})=\ell^2$; however, I don't really see it….



Orthonormal bases in RKHSs via interpolating sequences
It is true that if a sequence $(k\_n)$ is interpolating for $M(\mathcal{H})$, then the *normalized* Kernel vectors $g\_n:=K\_{k\_n}/\Vert K\_{k\_n} \Vert\_\mathcal{H} $ form a Riesz system in $\mathcal{H}$. Of course it does not have to be a complete system, so it is not always a Riesz basis.


To see this, let $g\_n$ be the normalized kernel vectors, and for $t\in \mathbb{R}, j\in \mathbb{N}$ define $w\_j = e^{2\pi i j t} $. Then by hypothesis, for every $t$ there exists a multiplier $\varphi\_t$ such that $\varphi\_t(z\_j) = w\_j,\, j=1,2,\dots$. Furthermore by the open mapping theorem we can choose $\varphi\_t$ such that $\Vert \varphi\_t \Vert\_{M(\mathcal{H})} \leq M < +\infty$. Then consider the adjoint of the multiplication operator by $\varphi\_t$, $M\_{\varphi\_t}^\*:\mathcal{H} \to \mathcal{H}$. By the bound on the multiplier norms of $\varphi\_t$,
\begin{equation}\label{eq1} \Vert M^\*\_{\varphi\_t} f \Vert\_\mathcal{H} ^2 \leq M^2 \Vert f \Vert\_\mathcal{H}^2,\,\,\, \forall t\in \mathbb{R}, \,\,\, \forall f \in \mathcal{H}. \end{equation}


Now consider $(a\_n)$ a sequence of complex numbers with only finite non zero terms, and $f=\sum\_{n}a\_n g\_n $. Applying the above inequality for this choice of $f$ and using the fact that $M^\*\_{\varphi\_t} g\_n = \overline{\varphi\_t(k\_n)} g\_n $ we arrive at the inequality,
$$ \sum\_{n,m} a\_n \overline{a\_m} e^{2\pi i (n-m) t} \langle g\_n , g\_m \rangle \leq M^2 \Vert \sum\_{n} a\_n g\_n \Vert^2, $$
and integrating with respect to $t$ in $(0,1)$ we find that
$$ \sum\_n |a\_n|^2 \leq M^2 \Vert \sum\_{n} a\_n g\_n \Vert^2 $$
By replacing $a\_n$ by $e^{-2\pi i tn} a\_n $ and using the same ineqaulity and integrating with respect to $t$ we find that
$$\Vert \sum\_{n} a\_n g\_n \Vert^2 \leq M^2 \sum\_n |a\_n|^2. $$
This proves that $g\_n$ is a Riesz system.


