Title: Switch Functions,
Abstract: We define a switch function to be a function from an interval to $\{1,-1\}$
with a finite number of sign changes. (Special cases are the Walsh functions.)
By a topological argument, we prove that, given $n$ real-valued functions,
$f_1, \dots, f_n$, in $L^1[0,1]$, there exists a switch function, $\sigma$,
with at most $n$ sign changes that is simultaneously orthogonal to all of them
in the sense that $\int_0^1 \sigma(t)f_i(t)dt=0$, for all $i = 1, \dots , n$.
Moreover, we prove that, for each $\lambda \in (-1,1)$, there exists a unique
switch function, $\sigma$, with $n$ switches such that $\int_0^1 \sigma(t) p(t)
dt = \lambda \int_0^1 p(t)dt$ for every real polynomial $p$ of degree at most
$n-1$. We also prove the same statement holds for every real even polynomial of
degree at most $2n-2$. Furthermore, for each of these latter results, we write
down, in terms of $\lambda$ and $n$, a degree $n$ polynomial whose roots are
the switch points of $\sigma$; we are thereby able to compute these switch
functions.