On A Class of Approximation Polygons for the Peano Curve Original Research Article

The Peano curve gives a continuous function of the unit interval onto the unit square respectively the unit cube. One of the most beautiful Peano carves was constructed in 1938 by I. J. Schoenberg. In this article we show how to use the theory of uniform distribution to construct as many points of this Peano curve as you need. In Section 3 we show how to use this for giving approximate solutions of equations. In Section 4 we develop a new theory of uniform distribution, using a singular measure, which can be applied to Brownian motion.