On the Fourier spectrum of functions on Boolean cubes

  title={On the Fourier spectrum of functions on Boolean cubes},
  author={A. Defant and M. Mastylo and A. P{\'e}rez},
  journal={Mathematische Annalen},
  • A. Defant, M. Mastylo, A. Pérez
  • Published 2017
  • Mathematics
  • Mathematische Annalen
  • Let f be a real-valued function of degree d defined on the n-dimensional Boolean cube $$\{ \pm 1\}^{n}$${±1}n, and $$f(x) = \sum _{S \subset \{1,\ldots ,n\}} \widehat{f}(S) \prod _{k \in S} x_k$$f(x)=∑S⊂{1,…,n}f^(S)∏k∈Sxk its Fourier-Walsh expansion. The main result states that there is an absolute constant $$C >0$$C>0 such that the $$\ell _{2d/(d+1)}$$ℓ2d/(d+1)-sum of the Fourier coefficients of $$f:\{ \pm 1\}^{n} \longrightarrow [-1,1]$$f:{±1}n⟶[-1,1] is bounded by $$C^{\sqrt{d \log d… CONTINUE READING
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