414/514 – The Schoenberg functions

Here is Jeremy Ryder’s project from last term, on the Schoenberg functions. Here we have a space-filling continuous map f:x\mapsto(\phi_s(x),\psi_s(x)) whose coordinate functions \phi_s and \psi_s are nowhere differentiable.

The proof that \phi_s,\psi_s are continuous uses the usual strategy, as the functions are given by a series to which Weierstrass M-test applies.

The proof that f is space filling is nice and short. The original argument can be downloaded here. A nice graph of the first few stages of the infinite fractal-like process that leads to the graph of f can be seen in page 49 of Thim’s master thesis.


One Response to 414/514 – The Schoenberg functions

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