The Universal Approximation Capabilities of 2pi-Periodic Approximate Identity Neural Networks

A fundamental theoretical aspect of artificial neural networks is related to the investigation of the universal approximation capability of a new type of a three-layer feed forward neural networks. In this study, we present four theorems concerning the universal approximation capabilities of a three-layer feed forward 2pi-periodic approximate identity neural networks. Using 2pi-periodic approximate identity, we prove two theorems which show the universal approximation capability of a three layer feed forward 2pi-periodic approximate identity neural networks in the space of continuous 2pi-periodic functions. The proofs of these theorems are based on the convolution linear operators and the theory of ε-net. Using 2pi-periodic approximate identity again, we also prove another two theorems which show the universal approximation capability of these networks in the space of pth-order Lebesgue integrable 2pi-periodic functions.