Moment Problems in Weighted L2 Spaces on the Real Line

Abstract

For a class of sets with multiple terms [FORMULA], having density d counting multiplicities, and a doubly-indexed sequence of non-zero complex numbers\linebr eak {dn,k:n∈N,k=0,1,…,μn−1} satisfying certain growth conditions, we consider a moment problem of the form [FORMULA], in weighted L2(−∞,∞) spaces. We obtain a solution f which extends analytically as an entire function, admitting a Taylor–Dirichlet series representation f(z)=∑n=1∞(∑k=0μn−1cn,kzk)eλnz,cn,k∈C,∀z∈C. The proof depends on our previous work where we characterized the closed span of the exponential system {tkeλnt:n∈N,k=0,1,2,…,μn−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.