Sigmoid functions and multiscale resolution of singularities

In this short, conceptual paper we observe that essentially the same mathematics applies in three contexts with disparate literatures: (1) sigmoidal and RBF approximation of smooth functions, (2) rational approximation of analytic functions near singularities, and (3) $hp$ mesh refinement for solution of PDEs. The relationship of (1) and (2) is as simple as the change of variables $s=\log (x)$, and our informal mnemonic for this relationship is ``sigmoid = log(ratapprox).‘’