The closed span of some Exponential system $E_{\Lambda }$ in the spaces $L^p(\gamma ,\beta )$, properties of a Biorthogonal family to $E_{\Lambda }$ in $L^2(\gamma ,\beta )$, Moment problems, and a differential equation of Carleson

Abstract

A set of complex numbers $\Lambda =\{{\lambda }_n,{\mu }_n{\}}_{n=1}^{\infty }$ with multiple terms \[ \{λ_n,μ_n\}_{n=1}^{\infty}:= \{\underbrace{λ_1,λ_1,\dots,λ_1}_{μ_1 - times}, \underbrace{λ_2,λ_2,\dots,λ_2}_{μ_2 - times},\dots, \underbrace{λ_k,λ_k,\dots,λ_k}_{μ_k - times},\dots\} \] is said to belong to the $\mathbf{ABC}$ class if it satisfies three conditions: $(\mathbf{A})$ ${\sum }_{n=1}^{\infty }{\mu }_n/|{\lambda }_n|<\infty$, $(\mathbf{B})$ ${\sup }_{n\in \mathbb{N}}|\arg {\lambda }_n|<\pi /2$, $(\mathbf{C})$ $\Lambda$ is an interpolating variety for the space of entire functions of exponential type zero. Assuming that $\Lambda \in \mathbf{ABC}$, we characterize in the spirit of the Müntz-Szász theorem, the closed span of its associated exponential system \[ E_Λ:=\{x^k e^{λ_n x}:\, n\in\mathbb{N},\,\, k=0,1,2,\dots,μ_n-1\} \] in the Banach spaces $L^p(\gamma ,\beta )$, where $-\infty <\gamma <\beta <\infty$ and $p\ge 1$. Related to $E_{\Lambda }$, we explore the properties of its unique biorthogonal sequence \[ r_Λ=\{r_{n,k}:\, n\in\mathbb{N},\, k=0,1,\dots,μ_n-1\}\subset\overline{\text{span}}(E_Λ) \] in $L^2(\gamma ,\beta )$. As a result, we find a solution to the Moment problem \[ \int_γ^β f(t)\cdot t^k e^{\overline{λ_n} t}\, dt=d_{n,k},\qquad \forall\,\, n\in\mathbb{N}\quad \text{and}\quad k=0,1,\dots ,μ_n-1,\quad d_{n,k}=O(e^{a\Reλ_n})\,\, for\,\, a<β. \] Finally, we characterize the solution space of a differential equation of infinite order, studied by L. Carleson.