Fourier representation

General

Every Boolean function [math]f:\{-1,1\}^n \to \{-1,1\}[/math] may be uniquely written as a multivariate polynomial:

[math] f(x) = \sum_{S \subseteq [n]} \hat{f}_S \prod_{i \in S} x_i [/math],

where [math]\hat{f}_S[/math] are real numbers called the Fourier coefficients of [math]f[/math].

The set of functions [math]\{\underset{i\in S}{\prod}x_{i}|S\subseteq [n] \}[/math] is an orthonormal basis of the vector space of all boolean functions [math]f:\{-1,1\}^{n}\to\mathbb{R}[/math] with the inner product:

[math]\left\lt f,g\right\gt =2^{-n}\underset{x\in\{-1,1\}^{n}}{\sum}f(x)g(x)=\underset{x\sim\{-1,1\}^{n}}{\mathbb{E}}[f(x)g(x)].[/math]

The Fourier Weight

For [math]f:\{-1,1\}^{n}\to\mathbb{R}[/math] and [math]0\leq k\leq n[/math] the Fourier weight of [math]f[/math] at degree [math]k[/math] (or depth [math]k[/math]) is:

[math]W^{k}[f]=\underset{S\subseteq\{1,...n\},|s|=k}{\sum\hat{f}(S)^{2}}[/math]

Some useful identities:

Parseval's identity: [math]\left\lt f,f\right\gt =\underset{S\in [n]}{\sum}\hat{f}(S)^{2}[/math]

Plancharel's identity: [math]\left\lt f,g\right\gt =\underset{S\in [n]}{\sum}\hat{f}(S)\hat{g}(S)[/math]

The convolution theorem: [math]\widehat{f * g}(S) = \widehat{f}(S) \widehat{g}(S)[/math], where [math](f*g)(x) = \mathbb{E}_{y}\left[f(y)g(x \circ y) \right][/math] (where [math]\circ[/math] is bit-wise multiplication).

Properties

• TODO. Start off with relation between Fourier weight and other complexity measures such as circuit size, sensitivity, decision tree depth.