Rademacher functions

Summary

The Rademacher functions $r_1, r_2, \dots$ on $[0, 1)$ are square waves that record the binary digits of a point. The value $r_k(t)$ is $1$ or $-1$ according as the $k$-th binary digit of $t$ is $0$ or $1$. They form an orthonormal system in $L^2([0, 1))$ which is not complete. For Lebesgue measure they are independent random signs, so they model coin tossing inside analysis. Their finite products are the Walsh functions.