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FKN Theorem on the biased cube

Piotr Nayar (2014)

Colloquium Mathematicae

We consider Boolean functions defined on the discrete cube - γ , γ - 1 equipped with a product probability measure μ n , where μ = β δ - γ + α δ γ - 1 and γ = √(α/β). This normalization ensures that the coordinate functions ( x i ) i = 1 , . . . , n are orthonormal in L ( - γ , γ - 1 , μ 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...

Fourier inversion of distributions on projective spaces

Francisco Javier González Vieli (2006)

Commentationes Mathematicae Universitatis Carolinae

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.

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