1
$\begingroup$

Let $p$ be some prime. Let $f \colon \mathbb{Z}_p \to \{0,1\}$ be some function, and define the function $g \colon \mathbb{Z}_p \to \{\pm 1\}$ as $g(x) = (-1)^{f(x)}$.

What can be said about Fourier spectrum of $f$ vs. $g$? Namely, does $\widehat{f}(\alpha)$ related to some $\widehat{g}(\beta)$ for some $\alpha, \beta \in \mathbb{Z}_p$?

I wanted to prove that they are equal in some sense, as the functions are just isomorphic to each other, but couldn't prove it yet.

As a more general perspective, take any two functions $f, g \colon \mathbb{Z}_p \to \mathbb{C}$ such that $f(x) = m(g(x))$, where $m \colon \mathbb{C} \to \mathbb{C}$ is some isomorphism. What can be said about $\widehat{f}(\alpha)$ vs. $\widehat{g}(\alpha)$ (for some $\alpha \in \mathbb{Z}_p$)?

Improve this question
$\endgroup$
4
  • 7
    $\begingroup$ $g=1-2f {}{} {}$ $\endgroup$
    – mathworker21
    Commented Jul 2, 2022 at 11:00
  • $\begingroup$ Thanks! That resolves my main question, as it implies $\widehat{g}(\alpha) = -2\widehat{f}(\alpha)$. I still not sure about the general case though $\endgroup$
    – Woka
    Commented Jul 2, 2022 at 12:15
  • 2
    $\begingroup$ What does "isomorphism" mean? $\endgroup$
    – mathworker21
    Commented Jul 2, 2022 at 13:57
  • $\begingroup$ Do you want to map to the complex unit circle? Maybe the $p$th roots of unity? $\endgroup$
    – kodlu
    Commented Jul 2, 2022 at 14:02

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .