Moment problems in weighted $L^2$ spaces on the real line

For a class of sets with multiple terms
$$ \{\lambda_n,\mu_n\}_{n=1}^{\infty}:=\{\underbrace{\lambda_1,\lambda_1,\dots,\lambda_1}_{\mu_1 - times}, \underbrace{\lambda_2,\lambda_2,\dots,\lambda_2}_{\mu_2 - times},\dots, \underbrace{\lambda_k,\lambda_k,\dots,\lambda_k}_{\mu_k - times},\dots\}, $$
having density $d$ counting multiplicities, and a doubly-indexed sequence of non-zero complex numbers\linebreak $\{d_{n,k}:\, n\in\mathbb{N},\, k=0,1,\dots ,\mu_n-1\} $ satisfying certain growth conditions, we consider a moment problem of the form
$$ \int_{-\infty}^{\infty}e^{-2w(t)}t^k e^{\lambda_n t}f(t)\, dt=d_{n,k},\quad \forall\,\, n\in\mathbb{N}\quad \text{and}\quad k=0,1,2,\dots, \mu_n-1, $$
in weighted $L^2 (-\infty, \infty)$ spaces. We obtain a solution $f$ which extends analytically as an entire function, admitting a Taylor-Dirichlet series representation
$$ f(z)=\sum_{n=1}^{\infty}\Big(\sum_{k=0}^{\mu_n-1}c_{n,k} z^k\Big) e^{\lambda_n z},\quad c_{n,k}\in \mathbb{C},\quad\forall\,\, z\in \mathbb{C}. $$
The proof depends on our previous work where we characterized the closed span of the exponential system $\{t^k e^{\lambda_n t}:\, n\in\mathbb{N},\,\, k=0,1,2,\dots,\mu_n-1\}$ in weighted $L^2 (-\infty, \infty)$ spaces, and also derived a sharp upper bound for the norm of elements of a biorthogonal sequence to the exponential system. The proof also utilizes notions from Non-Harmonic Fourier series such as Bessel and Riesz–Fischer sequences.