Simultaneous approximation by spherical neural networks

Approximation capabilities of the spherical neural networks (SNNs) are considered in this paper. Based on a known Taylor formula, we prove that, for non-polynomial target function, rates of simultaneously approximating the function itself and its (Laplace–Beltrami) derivatives by SNNs is not slower than those by the spherical polynomials (SPs). Then, the simultaneous approximation rates of SPs automatically derive the rates of SNNs.