Bohr's phenomenon for functions on the Boolean cube

Abstract

We study the asymptotic decay of the Fourier spectrum of real functions $f:\{-1,1{\}}^N\to \mathbb{R}$ in the spirit of Bohr's phenomenon from complex analysis. Every such function admits a canonical representation through its Fourier-Walsh expansion $f(x)={\sum }_{S\subset \{1,\dots ,N\}}\hat{f}(S)x^S,$ 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 $\rho \ge 0$ such that ${\sum }_S|\hat{f}(S)|{\rho }^{|S|}\le \Vert f{\Vert }_{\infty }$ 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.

26 pages