A characterization of interpolating sequences for multipliers of weighted Dirichlet spaces

Let $w_n$ be given by equation $(15)$, let ${\mathcal H}_{\mathbf w}$ be the corresponding Hilbert space, and let $Mult({\mathcal H}_{\mathbf w})$ denote its multiplier algebra. A sequence $\{\lambda_i\}_{i=1}^{\infty}$ is called interpolating for $Mult({\mathcal H}_{\mathbf w})$ when bounded multiplier data on the sequence can be interpolated by a multiplier. Characterization conjecture. The sequence $\{\lambda_i\}_{i=1}^{\infty}$ is interpolating for $Mult({\mathcal H}_{\mathbf w})$ if and only if both of the following conditions hold: (i) there exists a constant $C$ such that, for every $i\ne j$, there is a function $\phi\in Mult({\mathcal H}_{\mathbf w})$ with norm less than $C$ satisfying $\phi(\lambda_i)=0$ and $\phi(\lambda_j)=1$; and (ii) there exists a constant $C$ such that, for every $f\in {\mathcal H}_{\mathbf w}$,

$$\sum_{i=1}^{\infty}\frac{|f(\lambda_i)|^2}{\|k_{\lambda_i}\|^2}\leq C\|f\|_{{\mathcal H}_{\mathbf w}}^2.$$

The question follows a characterization of interpolation in terms of the normalized Gram matrix and the preceding theorem describing the maximal common domain of analyticity of the multipliers. Whether these two conditions characterize multiplier interpolation is left as an open problem in the source.