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The analysis connects spectral asymptotic asymptotics and lower bounds at the degrees where the ultraspherical coefficients do not vanish with Sobolev-norm approximation and coefficient control for fixed analytic activation functions.
This work constructs finite-dimensional de Rham subcomplexes generated by fixed-neuron shallow ReLU shallow ReLU neural networks and proves exactness in arbitrary dimension and provides a geometric sufficient condition for the required linear independence.
The results therefore place linearized neural-network approximation within the classical saturation framework and show that, although ReLU$^k$ network spaces can outperform finite elements of the same degree, this advantage is intrinsically limited.
It is proved that quasi-Chebyshev parameter sets with univariate resolution $m$ generate fixed feature spaces attaining the sharp $H^r$-to-to-H^s$ approximation order for a class of analytic activations satisfying a quantitative non-cancellation condition on their Taylor coefficients.
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