Function approximation with zonal function networks with activation functions analogous to the rectified linear unit functions

A zonal function (ZF) network on the $q$ dimensional sphere $\\mathbb{S}^q$ is\na network of the form $\\mathbf{x}\\mapsto \\sum_{k=1}^n\na_k\\phi(\\mathbf{x}\\cdot\\mathbf{x}_k)$ where $\\phi :[-1,1]\\to\\mathbf{R}$ is the\nactivation function, $\\mathbf{x}_k\\in\\mathbb{S}^q$ are the centers, and\n$a_k\\in\\mathbb{R}$. While the approximation properties of such networks are\nwell studied in the context of positive definite activation functions, recent\ninterest in deep and shallow networks motivate the study of activation\nfunctions of the form $\\phi(t)=|t|$, which are not positive definite. In this\npaper, we define an appropriate smoothess class and establish approximation\nproperties of such networks for functions in this class. The centers can be\nchosen independently of the target function, and the coefficients are linear\ncombinations of the training data. The constructions preserve rotational\nsymmetries.\n