MOMENT PROBLEMS IN WEIGHTED \(L^2\) SPACES ON THE REAL LINE - Elias Zikkos (Ural Mathematical Journal)
For a class of sets with multiple terms $$ \{λ_n,μ_n\}_{n=1}^{∞}:=\{\underbrace{λ_1,λ_1,…,λ_1}_{μ_1 - times}, \underbrace{λ_2,λ_2,…,λ_2}_{μ_2 - times},…, \underbrace{λ_k,λ_k,…,λ_k}_{μ_k - times},…\},$$having density \(d\) counting multiplicities, and a doubly-indexed sequence of non-zero complex numbers\linebr eak \(\{d_{n,k}:\, n\in\mathbb{N},\, k=0,1,… ,μ_n-1\} \) satisfying certain growth conditions, we consider a moment problem of the form $$\int_{-∞}^{∞}e^{-2w(t)}t^k e^{λ_n t}f(t)\, dt=d_{n,k},\quad \forall\,\, n\in\mathbb{N}\quad \text{and}\quad k=0,1,2,…, μ_n-1,$$ in weighted \(L^2 (-∞, ∞)\) spaces. We obtain a solution \(f\) which extends analytically as an entire function, admitting a Taylor–Dirichlet series representation $$ f(z)=\sum_{n=1}^{∞}\Big(\sum_{k=0}^{μ_n-1}c_{n,k} z^k\Big) e^{λ_n z},\quad c_{n,k}\in \mathbb{C},\quad\forall\,\, z\in \mathbb{C}. $$ The proof depends on our previous work where we characterized the closed span of the exponential system \(\{t^k e^{λ_n t}:\, n\in\mathbb{N},\,\, k=0,1,2,…,μ_n-1\}\) in weighted \(L^2 (-∞, ∞)\) 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.