Approximation of multivariate 2π-periodic functions by multiple 2π-periodic approximate identity neural networks based on the universal approximation theorems

Universal approximation capability is an important research topic in artificial neural networks. The purpose of this study is to investigate universal approximation capability of a single hidden layer feed forward multiple 2π-periodic approximate identity neural networks in two function spaces. We present the notion of multiple 2π-periodic approximate identity. With respect to this notion, we prove two theorems in the space of continuous multivariate 2π-periodic functions. The second theorem shows that the above networks have universal approximation capability. The proof of the theorem uses a technique based on the notion of epsilon-net. Moreover, we discuss the universal approximation capability of the networks in the space of Lebesgue integrable multivariate 2π-periodic functions. The results of this study will be able to extend the standard theory of the universal approximation capability of feedforward neural networks.