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.

Advertisements

One Response to 414/514 – The Schoenberg functions

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: