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A set of complex numbers $\Lambda=\{\lambda_n,\mu_n\}_{n=1}^{\infty}$ withmultiple terms \[ \{\lambda_n,\mu_n\}_{n=1}^{\infty}:=\{\underbrace{\lambda_1,\lambda_1,\dots,\lambda_1}_{\mu_1 - times},\underbrace{\lambda_2,\lambda_2,\dots,\lambda_2}_{\mu_2 - times},\dots,\underbrace{\lambda_k,\lambda_k,\dots,\lambda_k}_{\mu_k - times},\dots\} \] issaid 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 aninterpolating variety for the space of entire functions of exponential typezero. Assuming that $\Lambda\in\bf ABC$, we characterize in the spirit of theM\"{u}ntz-Sz\'{a}sz theorem, the closed span of its associated exponentialsystem \[ E_{\Lambda}:=\{x^k e^{\lambda_n x}:\, n\in\mathbb{N},\,\,k=0,1,2,\dots,\mu_n-1\} \] in the Banach spaces $L^p(\gamma,\beta)$, where$-\infty<\gamma<\beta<\infty$ and $p\ge 1$. Related to $E_{\Lambda}$, weexplore 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, wecharacterize the solution space of a differential equation of infinite order,studied by L. Carleson.

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