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I don't see how this improves upon the Stone-Weierstrass theorem, aside from expanding the range of functions that can uniformly approximate a continuous function or scatter plot. A scatter plot can be turned into a continuous function by setting the value to be the linear function that connects the two nearest data points, a piecewise linear function.

https://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theo...

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The improvement is that the function can represent any plot by adjusting a single coefficient, whereas a polynomial approximation has a number of coefficients equal to the number of points in the plot.

This is only an improvement in a superficial, linguistic sense though. If the single coefficient is just a bit-packing representation of many more degrees of freedom (because of its huge precision), then from a model information complexity point of view, the polynomial model could actually have fewer parameters, in the sense that the overall size of the combined parameter space is smaller, e.g. it’s a smaller program size.

It reminds me of the Grue vs. Bleen question in philosophy.

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