Uniform Approximation of Functions with Asymmetric Growth and Decay by Deep Weighted Polynomials
Functions that grow without bound on one side of the real line and decay to zero on the other cannot be approximated uniformly by ordinary polynomials on unbounded domains. Motivated by classical weighted polynomial approximation, we introduce a class of one-sided weighted \emph{deep} (composite) polynomial approximants for such asymmetric targets. The weight suppresses polynomial growth on the decaying side, while the composite polynomial remains free to capture growth on the other side. We prove that this mechanism reduces the half-line approximation problem to approximation on a compact interval whose length grows slowly with the degree, and we establish density and existence of best approximants in the appropriate closure of the model class. For computation, we first formulate the method as a trainable computational graph for \emph{deep} weighted polynomial approximation. However, direct end-to-end optimization becomes increasingly ill-conditioned at high composite degree and can suffer from local minima. To address this, we introduce a fine-tuning procedure in which a fixed inner composition of monotone polynomial self-maps supplies the effective degree, while only the outer polynomial and weight parameters are trained; the outer fit reduces to a linear program. Numerical experiments on Black--Scholes option-pricing functions show that the resulting fine-tuned weighted \emph{deep} polynomial achieves smaller uniform and \(L_2\) errors than matched-budget polynomial baselines and resolves the decaying tail to machine precision.