Representation of a crinkled arc

Johnson [A crinkled arc, Proc. Amer. Math. Soc. 25 (1970), 375-376] has shown that under suitable normalizations all crinkled arcs are unitarily equivalent. Using this result, we find a general series expansion for a crinkled arc:

$$f(t) = \sqrt 2 \sum\limits_{n = 1}^\infty {{x_n}\tfrac{{\sin (n -... ...yle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}})\pi }}} ,$$

where $\{ {x_n}\}$ is an orthonormal set.