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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 incr","title":"Uniform Approximation of Functions with Asymmetric Growth and Decay by Deep Weighted Polynomials","url":"https://arxiv.org/abs/2506.21306","vendor":"arxiv_cs_ai"},"summary":"arXiv:2506.21306v2 Announce Type: replace-cross \nAbstract: 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. 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