<![CDATA[Function approximation lies at the boundary between classical analysis and modern computing, although the literature tends to consider its classical and modern branches as independent. In this paper, we propose that polynomial neural networks with a single hidden layer, methods, and Hilbert space projections are not only identical but also share a fundamental structure that can be explicitly expressed. We revisit Weiserstrasse and Bernstein's probabilistic construction theories, develop a Hilbert space projection framework, and investigate numerical problems such as node positioning and least-squares stability. All these lines converge in a discussion of Sybenko's theory of total approximation, which can be read as a classical result rather than a break from it. Numerical experiments on the Fourier square wave reconstruction and the Fourier square wave reconstruction are accompanied by theoretical development on . There is an error and stability at which the paper concludes. The analysis can be applied to all three approximation models.]]>
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