Very interesting! Kolmogorov neutral networks can represent discontinuous functions [1], but I've wondered about how practically applicable they are. This repo seems to show that they have some use after all.
Not for discontinuous functions, as your paper explains, we know that g exists for discontinuous bounded, but nothing to find it with.
> A practical construction of g in cases with discontinuous bounded and un-
bounded functions is not yet known. For such cases Theorem 2.1 gives only a theoretical understanding of the representation problem. This is because for the representation of discontinuous bounded functions we have derived (2.1) from the fact that the range of the operator Z∗ is the whole space of bounded functions B(Id). This fact directly gives us a formula (2.1) but does not tell how the bounded one-variable function g is attained. For the representation of unbounded functions we have used a linear extension of the functional F , existence of which is based on Zorn’s lemma (see, e.g., [19, Ch. 3]). Application of Zorn’s lemma provides no mechanism for practical construction of such an extension. Zorn’s lemma helps to assert only its existence.
If you look at the OP post arxiv link, you will see they are using splines .
[1]: https://arxiv.org/abs/2311.00049