While attention for and usage of Deep Neural Network (DNN) based applications skyrocket, the mathematical understanding of their behavior and capabilities is still in its infancy. Contrary to traditional approaches, that depend on training by loss minimization algorithms, a method will be presented to explicitly construct DNNs that emulate multivariate Chebyshev polynomials and can be used to approximate a large class of functions. The theory of this method, it's accuracy and it's bounds on depth and size will be introduced as well as an implementation and comparison to training-based DNNs.