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, f1,…,fn, in L1[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 ∫σ(t)fi(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 ∫σ(t)p(t)dt=λ∫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.
Moreover, we prove that, for each λ∈(−1,1), there exists a unique switch function, σ, with n switches such that ∫σ(t)p(t)dt=λ∫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.
| Original language | English |
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| Publisher | arXiv |
| Number of pages | 26 |
| Publication status | Published - 8 Oct 2017 |