An optimal uncertainty principle in twelve dimensions via modular forms
An optimal uncertainty principle in twelve dimensions via modular forms
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通过模块化形式的十二维最优不确定性原理
DOI：
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发表时间：
2017
影响因子：
3.1
通讯作者：
Felipe Gonçalves
中科院分区：
文献类型：
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作者：
Henry Cohn;Felipe Gonçalves
我们证明了Bourain、Clozel和Kahane的测不准原理在十二个维度上的最优界。假设$$f：mathbb{R}^{12}
右Mathbb{R}$$f：R12→R是不恒为零的可积函数。将其傅里叶变换$$widehat{f}$$f^归一化为$$widehat{f}(Xi)=int_{mathbb{R}^d}f(X)e^{-2pi i langx，xi
ξ)=∫}，dx$$f^(Angel rdf(X)e-2πi⟨x，ξ⟩dx，并假设$$widehat{f}$$f^是实值可积的。证明了如果$$f(0)le 0$$f(0)≤0，$$widehat{f}(0)le 0$$f^(0)≤0，$$f(X)ge0$$f(X)≥0对于$$|x|ge r_1$$|x|≥r1，以及$$widehat{f}(Xi)ge 0$$f^(ξ)≥0表示$$|xi|ge r_2$$|ξ|≥R2)，则$$r_1R_2ge 2$$r1r2≥2，并且这个界是尖锐的。函数的构造是基于Viazovska的模形式技巧，它的最优性源于Eisenstein级数$$E6$$E6的存在。在任何其他维度上，都没有已知的尖锐界限，甚至没有人猜测。我们还发展了与Cohn和Elkies的线性规划界之间的联系，从而推广了f和$$widehat{f}$$f^的符号模式，从而发展了互补的测不准原理。这一推广将不确定原理与线性规划界限统一为更广泛的理论的各个方面。
We prove an optimal bound in twelve dimensions for the uncertainty principle of Bourgain, Clozel, and Kahane. Suppose $$f :mathbb {R}^{12}
ightarrow mathbb {R}$$f:R12→R is an integrable function that is not identically zero. Normalize its Fourier transform $$widehat{f}$$f^ by $$widehat{f}(xi ) = int _{mathbb {R}^d} f(x)e^{-2pi i langle x, xi
angle }, dx$$f^(ξ)=∫Rdf(x)e-2πi⟨x,ξ⟩dx, and suppose $$widehat{f}$$f^ is real-valued and integrable. We show that if $$f(0) le 0$$f(0)≤0, $$widehat{f}(0) le 0$$f^(0)≤0, $$f(x) ge 0$$f(x)≥0 for $$|x| ge r_1$$|x|≥r1, and $$widehat{f}(xi ) ge 0$$f^(ξ)≥0 for $$|xi | ge r_2$$|ξ|≥r2, then $$r_1r_2 ge 2$$r1r2≥2, and this bound is sharp. The construction of a function attaining the bound is based on Viazovska’s modular form techniques, and its optimality follows from the existence of the Eisenstein series $$E_6$$E6. No sharp bound is known, or even conjectured, in any other dimension. We also develop a connection with the linear programming bound of Cohn and Elkies, which lets us generalize the sign pattern of f and $$widehat{f}$$f^ to develop a complementary uncertainty principle. This generalization unites the uncertainty principle with the linear programming bound as aspects of a broader theory.