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Let $$ f_{\alpha}(x)=\phi_1(\alpha) \mathrm{e}^{-\frac{|x|^\alpha}{\phi_2(\alpha) }}, x \in \mathbb{R}, 0<\alpha<2, $$ where $\phi_1(\alpha)=\frac{\alpha}{2} \left\{{\{\Gamma(3/\alpha)\}^{1/...

Consider choosing a Boolean function $f : \{0, 1\}^{n} \rightarrow \{-1, 1\}$ uniformly at random from the set of all Boolean functions and consider the random variable $\left(\hat f(z_{1}), \hat f(z_{...

Short version of question: I'm trying to understand the extent to which a function is prevented from being "flat" as a result of being the Fourier transform of a positive function. That is, the extent ...