The Fourier Transform of Restrictions of Functions on the Slice

TL;DR

This paper explores the Fourier transform of functions on the slice of the Boolean hypercube, establishing relationships between Fourier coefficients of functions and their restrictions, and applying this to develop a Goldreich-Levin theorem for the slice.

Contribution

It introduces a relationship between Fourier coefficients of functions on the slice and their restrictions, and extends the Goldreich-Levin theorem to this setting.

Findings

Established a relationship between Fourier coefficients of functions on the slice and their restrictions.

Proved a Goldreich-Levin theorem for functions on the slice.

Extended the Kushilevitz-Mansour algorithm to the slice setting.

Abstract

This paper considers the Fourier transform over the slice of the Boolean hypercube. We prove a relationship between the Fourier coefficients of a function over the slice, and the Fourier coefficients of its restrictions. As an application, we prove a Goldreich-Levin theorem for functions on the slice based on the Kushilevitz-Mansour algorithm for the Boolean hypercube.