On the Fourier spectrum of functions on Boolean cubes
On the Fourier spectrum of functions on Boolean cubes
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布尔立方体上函数的傅立叶谱
DOI：
10.1007/s00208-018-1756-y
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发表时间：
2017
影响因子：
1.4
通讯作者：
A. Pérez
中科院分区：
文献类型：
--
作者：
A. Defant;M. Mastyło;A. Pérez
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 $$C^{\sqrt{d \log d}}$$Cdlogd. It was recently proved that a similar result holds for complex-valued polynomials on the n-dimensional polytorus $$\mathbb {T}^n$$Tn, but that in contrast to this, a replacement of the n-dimensional torus $$\mathbb {T}^n$$Tn by the n-dimensional cube $$[-1, 1]^n$$[-1,1]n leads to a substantially weaker estimate. This in the Boolean case forces us to invent novel techniques which differ from the ones used in the complex or real case. We indicate how our result is linked with several questions in quantum information theory.
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 $$C^{\sqrt{d \log d}}$$Cdlogd. It was recently proved that a similar result holds for complex-valued polynomials on the n-dimensional polytorus $$\mathbb {T}^n$$Tn, but that in contrast to this, a replacement of the n-dimensional torus $$\mathbb {T}^n$$Tn by the n-dimensional cube $$[-1, 1]^n$$[-1,1]n leads to a substantially weaker estimate. This in the Boolean case forces us to invent novel techniques which differ from the ones used in the complex or real case. We indicate how our result is linked with several questions in quantum information theory.
影响因子：
1.3
作者：
Montanaro A
通讯作者：
Montanaro A