Spherical approximate identity neural networks are universal approximators

Approximation continuous functions on the unit sphere has important applications in science and engineering. The aim of this study is to answer questions concerning the universal approximation capability of a single-hidden layer feedforward spherical approximate identity neural networks to continuous functions on the unit sphere. First, the basic definitions of spherical convolution is introduced. Then, an obtained theorem shows that the convolution linear operators of spherical approximate identity with every continuous function / on the unit sphere converges to /. Making use of this result, a main theorem is also obtained. The method is used to prove the main theorem which is based on the theory of e-net. The results shows that spherical approximate identity neural networks are universal approximators.