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 \[ \{\lambda_n,\mu_n\}_{n=1}^{\infty}:= \{\underbrace{\lambda_1,\lambda_1,\dots,\lambda_1}_{\mu_1 - times}, \underbrace{\lambda_2,\lambda_2,\dots,\lambda_2}_{\mu_2 - times},\dots, \underbrace{\lambda_k,\lambda_k,\dots,\lambda_k}_{\mu_k - times},\dots\} \] is said to belong to the $\bf ABC$ class if it satisfies three conditions: $\bf (A)$ $\sum_{n=1}^{\infty}\mu_n/|\lambda_n|<\infty$, $\bf (B)$ $\sup_{n\in\mathbb{N}}|\arg\lambda_n|<\pi/2$, $\bf (C)$ $\Lambda$ is an interpolating variety for the space of entire functions of exponential type zero. Assuming that $\Lambda\in\bf ABC$, we characterize in the spirit of the M\"{u}ntz-Sz\'{a}sz theorem, the closed span of its associated exponential system \[ E_{\Lambda}:=\{x^k e^{\lambda_n x}:\, n\in\mathbb{N},\,\, k=0,1,2,\dots,\mu_n-1\} \] 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 \[ r_{\Lambda}=\{r_{n,k}:\, n\in\mathbb{N},\, k=0,1,\dots,\mu_n-1\}\subset\overline{\text{span}}(E_{\Lambda}) \] in $L^2(\gamma,\beta)$. As a result, we find a solution to the Moment problem \[ \int_{\gamma}^{\beta} f(t)\cdot t^k e^{\overline{\lambda_n} t}\, dt=d_{n,k},\qquad \forall\,\, n\in\mathbb{N}\quad \text{and}\quad k=0,1,\dots ,\mu_n-1,\quad d_{n,k}=O(e^{a\Re\lambda_n})\,\, for\,\, a<\beta. \] Finally, we characterize the solution space of a differential equation of infinite order, studied by L. Carleson.