Boolean functions whose Fourier transform is concentrated on the first two levels

Authors:
Ehud Friedgut;Gil Kalai;Assaf Naor
Affiliations:
Institute of Mathematics, Hebrew University, Jerusalem, Israel;Institute of Mathematics, Hebrew University, Jerusalem, Israel;Institute of Mathematics, Hebrew University, Jerusalem, Israel
Venue:
Advances in Applied Mathematics
Year:
2002

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Abstract

In this note we describe Boolean functions f(x"1,x"2,...,x"n) whose Fourier coefficients are concentrated on the lowest two levels. We show that such a function is close to a constant function or to a function of the form f=x"k or f=1-x"k. This result implies a ''stability'' version of a classical discrete isoperimetric result and has an application in the study of neutral social choice functions. The proofs touch on interesting harmonic analysis issues.