Bohr's phenomenon for functions on the Boolean cube

Abstract

We study the asymptotic decay of the Fourier spectrum of real functions $f\colon \{-1,1\}^N \rightarrow \mathbb{R}$ in the spirit of Bohr's phenomenon from complex analysis. Every such function admits a canonical representation through its Fourier-Walsh expansion $f(x) = \sum_{S\subset \{1,\ldots,N\}}\widehat{f}(S) x^S \,,$ where $x^S = \prod_{k \in S} x_k$. Given a class $\mathcal{F}$ of functions on the Boolean cube $\{-1, 1\}^{N} $, the Boolean radius of $\mathcal{F}$ is defined to be the largest $\rho \geq 0$ such that $\sum_{S}{|\widehat{f}(S)| \rho^{|S|}} \leq \|f\|_{\infty}$ for every $f \in \mathcal{F}$. We give the precise asymptotic behaviour of the Boolean radius of several natural subclasses of functions on finite Boolean cubes, as e.g. the class of all real functions on $\{-1, 1\}^{N}$, the subclass made of all homogeneous functions or certain threshold functions. Compared with the classical complex situation subtle differences as well as striking parallels occur.