Moment problems in weighted L² spaces on the real line

For a class of sets with multiple terms (formula presented) having density d counting multiplicities, and a doubly-indexed sequence of non-zero complex numbers {d_n,k: n ∈ N, k = 0, 1, …, µ_n − 1} satisfying certain growth conditions, we consider a moment problem of the form (formula presented) in weighted L² (−∞, ∞) spaces. We obtain a solution f which extends analytically as an entire function, admitting a Taylor-Dirichlet series representation (formula presented) The proof depends on our previous work where we characterized the closed span of the exponential system {t^k e^λnt: n ∈ N, k = 0, 1, 2, …, µ_n −1} in weighted L² (−∞, ∞) 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.