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Israel Journal of Mathematics

, Volume 131, Issue 1, pp 269–276 | Cite as

On the distribution of the fourier spectrum of Boolean functions

  • J. Bourgain
Article

Abstract

In this paper we obtain a general lower bound for the tail distribution of the Fourier spectrum of Boolean functionsf on {1, −1} N . Roughly speaking, fixingk∈ℤ+ and assuming thatf is not essentially determined by a bounded number (depending onk) of variables, we have that\(\sum {\left| s \right| > k\left| {\hat f(S)} \right|^2 } \gtrsim k^{ - 1/2 - \varepsilon } \). The example of the majority function shows that this result is basically optimal.

Keywords

Boolean Function American Mathematical Society Complexity Theory FOURIER Spectrum Fourier Expansion 
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References

  1. [Fr]
    E. Friedgut,Sharp threshold of graph properties, and the k-sat problem, Journal of the American Mathematical Society12 (1999), 1017–1054.MATHCrossRefMathSciNetGoogle Scholar
  2. [KKL]
    J. Kahn, G. Kalai and N. Linial,The influence of variables on Boolean functions, Proc. 29th IEEE FOCS 58-80, IEEE, New York, 1988.Google Scholar
  3. [K]
    M. G. Karpovsky,Finite Orthogonal Series in the Design of Digital Devices, Wiley, New York, 1976.MATHGoogle Scholar

Copyright information

© The Hebrew University Magnes Press 2002

Authors and Affiliations

  1. 1.School of MathematicsInstitute for Advanced StudyPrincetonUSA

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