We consider Boolean functions defined on the discrete cube $-\gamma ,{\gamma }^{-1}ⁿ$ equipped with a product probability measure ${\mu }^{\otimes n}$, where $\mu =\beta {\delta }_{-\gamma }+\alpha {\delta }_{{\gamma }^{-1}}$ and γ = √(α/β). This normalization ensures that the coordinate functions ${\left(x_i\right)}_{i=1,...,n}$ are orthonormal in $L₂(-\gamma ,{\gamma }^{-1}ⁿ,{\mu }^{\otimes n})$. We prove that if the spectrum of a Boolean function is concentrated on the first two Fourier levels, then the function is close to a certain function of one variable. Our theorem strengthens the non-symmetric FKN Theorem due to Jendrej, Oleszkiewicz and Wojtaszczyk. Moreover, in the symmetric...

We show that the Fourier-Laplace series of a distribution on the real, complex or quarternionic projective space is uniformly Cesàro-summable to zero on a neighbourhood of a point if and only if this point does not belong to the support of the distribution.