Kolmogorov's superposition theorem states that if $f:[0,1]^n \to \mathbb{R}$ is an arbitrary continuous function, then it has the representation \begin{align} f(x) = \sum_{q=0}^{2n} \Phi_q (\sum_{p=1}^n \psi_{q,p}(x_p)) \end{align} where $\Phi_q,\psi_{q,p}$ are continuous one-dimensional functions.

While the functions $\psi_{q,p}(x_p)$ can be chosen to have ``nice properties'' (such as being Lipschitz continuous), the function $\Phi_q$ has no such guarantees.

Can we say something about $\Phi_q$ if $f$ is restricted to be Lipschitz continuous?



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