Asymptotic insights for projection, Gordon-Lewis and Sidon constants in Boolean cube function spaces

TL;DR

This paper investigates the asymptotic behavior of projection, Sidon, and Gordon-Lewis constants in Boolean cube function spaces, providing exact formulas and estimates that deepen understanding of their interrelations.

Contribution

It introduces new asymptotic formulas and estimates for key Banach space constants in Boolean cube function spaces, linking these constants through local Banach space theory.

Findings

Derived exact formulas for constants in specific families

Provided asymptotic estimates depending on support set complexity

Established relationships among the constants using local Banach space theory

Abstract

The main aim of this work is to study important local Banach space constants for Boolean cube function spaces. Specifically, we focus on $\mathcal{B}_{\mathcal{S}}^N$, the finite-dimensional Banach space of all real-valued functions defined on the $N$-dimensional Boolean cube $\{-1, +1\}^N$ that have Fourier--Walsh expansions supported on a fixed~family $\mathcal{S}$ of subsets of $\{1, \ldots, N\}$. Our investigation centers on the projection, Sidon and Gordon--Lewis constants of this function space. We combine tools from different areas to derive exact formulas and asymptotic estimates of these parameters for special types of families $\mathcal{S}$ depending on the dimension $N$ of the Boolean cube and other complexity characteristics of the support set $\mathcal{S}$. Using local Banach space theory, we establish the intimate relationship among these three important constants.