Switch Functions

TL;DR

This paper introduces the concept of switch functions, proves their existence and uniqueness under certain conditions, and provides explicit methods to compute them, with applications to orthogonality and polynomial approximation.

Contribution

It establishes the existence and uniqueness of switch functions orthogonal to given functions and characterizes them explicitly for polynomial cases, extending previous understanding.

Findings

Existence of switch functions orthogonal to given functions.

Uniqueness of switch functions with specified integral properties.

Explicit polynomial formulas for switch points.

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.