复变函数逼近论是函数论中的一个重要分支,与实变函数逼近论一样, 既有广泛的实际背景,也有很多值得研究的理论问题.复变函数逼近论的历史最早可追溯到1885年的~Runge~定理,后来~Bernstein、M\"{u}ntz、Malliavin~等许多数学家继续了复变函数逼近论的研究,考虑了幂函数系或指数函数系在不同函数空间,各种权函数情况下的完备性问题. 近年来,~Seip、Chistyakov~等人结合概率论和函数论的方法,把指数系的经典问题进行概率推广, 拓宽了新的视野并得到了一些新的结果.受他们文章的启发, 在本文中,我们主要研究了指数函数系和随机指数函数系在三个不同函数空间中的完备性问题. 首先, 结合概率论和函数论的方法, 我们研究了在不同条件下, 随机指数系~$\mathcal{E} (\Lambda_\omega) = \{t^l \exp ( \lambda_n(\omega)t)\}_{l=0, \ \ n\in\mathbb{N}_+}^{m_n(\omega)-1}$~在~Banach~空间~$C_\alpha$~中的完备性, 得到了随机指数系~$\mathcal{E}(\Lambda_\omega)$~在空间~$C_\alpha$~中在加权一致范数下以概率1完备和极小的充分必要条件,其中~$C_\alpha$~是由实数轴上连续函数组成的带凸权的空间.我们还证明了在随机指数函数系~$\mathcal{E}(\Lambda_\omega)$~不以概率1完备的情况下, 对几乎所有的~$\omega$,其闭包~$\overline{\mbox{span}} \mathcal{E}(\Lambda_\omega)$~中的任意元素都可以延拓为由~Taylor-Dirichlet~表示的整函数.这就是所谓的~Malliavin~经典定理关于指数系在~Banach~空间中的完备性的概率推广. 其次, 我们得到了指数函数系~$\{z^l \exp (-\lambda_n z)\}_{l=0, \ \ n\in\mathbb{N}_+}^{m_n-1}$~在空间~$H_\alpha$~中在一致范数下完备的充分必要条件,以及随机指数函数系~$\{z^l \exp ( -\lambda_n(\omega)z)\}_{l=0, \ \ n\in\mathbb{N}_+}^{m_n-1}$~在空间$H_\alpha$~中在一致范数下以概率1完备的充分必要条件,其中~$H_\alpha$~是由闭右半带形~$I_\alpha$~上满足在~$I_\alpha$~内部解析且在无穷远点为零的连续函数组成的空间. 最后, 我们推广了~Vinnitskii~的结果, 考虑了分别当~$1 ﹀

Approximation of complex functions is an important branch of the function theory. Like the approximation of real functions, on the one hand, it is widely used in extensive practical background; on the other hand, there are many theoretical problems to be studied. The history of approximation of complex functions dated back to Runge theorem in 1885, Bernstein, M\"{u}ntz, Malliavin and many othermathematicians went on studying the approximation of complex functions, and considered the completeness of the power function system or exponential system in different function spaces, in a variety of weight functions. In recent years, combining the methods of probability theory and function theory, Seip, Chistyakov and others, generalized the classical questions on exponential systems and obtained the probabilistic generalization of Malliavin's classical results, that gave a new insight and led to new results. Motivated by their works, in this dissertation, we mainly study the completeness of exponential systems and random exponential systems in three different function spaces. First, combining the methods of probability theory and function theory, we obtain necessary and sufficient conditions for the completeness of a complex exponential system $\mathcal{E}(\Lambda_\omega) = \{t^l \exp ( \lambda_n(\omega)t) \}_{l=0, \ \ n\in\mathbb{N}_+}^{m_n(\omega)-1}$ in the weighted Banach space $C_\alpha$ with a weighted uniform norm, in different conditions, where $C_\alpha$ is the space consisting of functionscontinuous on the real line for a convex weight. Moreover, we describe the closure of the incomplete system as follows: each function in the closure $\overline{\mbox{span}} \mathcal{E}(\Lambda_\omega)$ can be extended to an entire function represented by a Taylor-Dirichlet series. The results are viewed as a probabilistic generalization of Malliavin's classical results. Second, we obtain necessary and sufficient conditions for the completeness of exponential system $\{z^l \exp ( -\lambda_n z)\}_{l=0, \ \ n\in\mathbb{N}_+}^{m_n-1}$ in the Banach space $H_\alpha$,with uniform norm,consisting of functions continuous on the closed right half strip,analytic in its interior and vanishing uniformly at infinity.In addition, we obtain necessary and sufficient conditions for the completeness of random exponential system$\{z^l \exp (-\lambda_n(\omega)z)\}_{l=0, \ \n\in\mathbb{N}_+}^{m_n(\omega)-1}$ with probability 1. Finally, we generalize Vinnitskii's results, and consider the completeness and minimality of exponential system $E (\Lambda)=\{z^l\exp(\lambda_n z)\}_{l=0, \ \ n\in\mathbb{N}_+}^{m_n-1}$ in the space $E^p[\sigma]$ with $L^p$ norm, where the space $E^p[\sigma]$ is the class consisting of $L^p$ integrable functions analytic in half-strip $D_\sigma$ when $1 ﹀