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Switch Functions

Profile image of Bernard KayBernard Kay

2017, arXiv (Cornell University)

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 ,. .. , f n , in L 1 [0, 1], there exists a switch function, σ, with at most n sign changes that is simultaneously orthogonal to all of them in the sense that 1 0 σ(t)f i (t)dt = 0, for all i = 1,. .. , n. Moreover, we prove that, for each λ ∈ (−1, 1), there exists a unique switch function, σ, with n switches such that 1 0 σ(t)p(t)dt = λ 1 0 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 λ and n, a degree n polynomial whose roots are the switch points of σ; we are thereby able to compute these switch functions.

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