Lecture Details

Name: Andreas Defant (University of Oldenburg)

Speaking: Thu 9th May 09:30 - 09:55

Title: Bohr's phenomenon for functions on the Boolean cube

Abstract: In their two articles from 1989 and 1991 Sean Dineen and Richard Timoney initiated the study of multidimensional Bohr radii for holomorphic functions on the $N$-dimensional disc $\mathbb{D}^N$. In our talk we discuss similar phenomena for real-valued functions $f$ on the Boolean cube $\{-1,1\}^N$. 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 $\{-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 behavior of the Boolean radius of several natural classes of functions on Boolean cubes. Compared with the classical complex situation subtle differences as well as striking parallels occur.
Joint work with M. Mastyło and A. Pérez.

Lecture Slides