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

Published 1 Feb 2023 in math.FA and math.MG | (2302.00233v2)

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.

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