### Abstract

We consider Boolean functions defined on the discrete cube {-γ, γ^{-1}}^{n} equipped with a product probability measure µ^{⊗n}, where µ = βδ-γ + αδγ-1 and γ = α/β. This normalization ensures that the coordinate functions (xi)i=1,…,n are orthonor-mal in L2({-γ, γ^{-1}}^{n}, µ^{⊗n}). We prove that if the spectrum of a Boolean function is con-centrated 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 case α = β = 1/2 we prove that if a [-1, 1]-valued function defined on the discrete cube is close to a certain affine function, then it is also close to a [-1, 1]-valued affine function.

Original language | English (US) |
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Pages (from-to) | 253-261 |

Number of pages | 9 |

Journal | Colloquium Mathematicum |

Volume | 137 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 2014 |

### Keywords

- Boolean functions
- FKN theorem
- Walsh-fourier expansion

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## Cite this

*Colloquium Mathematicum*,

*137*(2), 253-261. https://doi.org/10.4064/cm137-2-9