In this work, we consider the weight spaces of integrable functions $L_p^\omega$ ($p\geq 1$) and continuous functions $C^\omega$ on the real line. Let $\Lambda=\{\lambda_k,n_k\}$ be an unbounded increasing sequence of positive numbers $\lambda_k$ and their multiplicities $n_k$, $\mathcal{E}(\Lambda)=\{t^n e^{\lambda_k t}\}$ be a system of exponential monomials constructed from the sequence $\Lambda$.
