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24 June 2024
  » arxiv » 2302.00233

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Asymptotic estimates of projection and Sidon constants for spaces of functions on the Boolean cube
Andreas Defant ; Daniel Galicer ; Martín Mansilla ; Mieczysław Mastyło ; Santiago Muro ;
Date 1 Feb 2023
AbstractWe investigate the projection constant $oldsymbol{lambda}ig(mathcal{B}_{mathcal{S}}^Nig)$ of the finite-dimensional Banach space $mathcal{B}_{mathcal{S}}^N$ of all real-valued functions defined on the $N$-dimensional Boolean cube ${-1, +1}^N$ that have Fourier-Walsh expansions supported on a~family $mathcal{S}$ of subsets of ${1, ldots, N}$. We combine ideas and tools from Fourier analysis, combinatorics, probability, and number theory to derive exact formulas and asymptotic estimates of $oldsymbol{lambda}ig(mathcal{B}_{mathcal{S}}^Nig)$ for special types of families $mathcal{S}$ depending on the dimension $N$ of the Boolean cube and other complexity parameters of $mathcal{S}$. One of the main results states that if $mathcal{S}= {S:, ext{card}(S) = d}$ or $mathcal{S}= {S: , ext{card}(S) leq d}$, then $N^{-d/2} oldsymbol{lambda}ig(mathcal{B}_{mathcal{S}}^Nig) o frac{1}{2pi} int_{mathbb{R}} |P_d(t)| exp(-t^2/2),d!t$ as $N o infty$, where $P_d$ is a concrete real polynomial of one variable. Using local Banach space theory, we show that the Sidon, Gordon-Lewis and the projection constants of each Banach space $mathcal{B}_{mathcal{ S}}^N$ are closely related.
Source arXiv, 2302.00233
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