In this paper, we extend several approximation theorems, originally formulated in the context of the standard \(L^p\) norm, to the more general framework of variable exponent spaces. Our study is motivated by applications in neural networks, where function approximation plays a crucial role. In addition to these generalizations, we provide alternative proofs for certain well-known results concerning the universal approximation property. In particular, we highlight spaces with variable exponents as illustrative examples, demonstrating the broader applicability of our approach. The paper is self-contained, although the paper reviews many well-known results. By the use of Fourier analysis, new proofs are given.
