## Abstract

In this paper we consider bounded real-valued functions over the discrete cube, f: {-1, 1}^{n} → [-1, 1]. Such functions arise naturally in theoretical computer science, combinatorics, and the theory of social choice. It is often interesting to understand when these functions essentially depend on few coordinates. Our main result is a dichotomy that includes a lower bound on how fast the Fourier coefficients of such functions can decay: we show that ∑ {|S| > f(S)^{2} exp( - O(k^{2} logk))}, unless f depends essentially only on 2 ^{O(k)} coordinates. We also show, perhaps surprisingly, that this result is sharp up to the log k factor. p ]The same type of result has already been proven (and shown useful) for Boolean functions [Bou02, KS]. The proof of these results relies on the Booleanity of the functions, and does not generalize to all bounded functions. In this work we handle all bounded functions, at the price of a much faster tail decay. As already mentioned, this rate of decay is shown to be both roughly necessary and sufficient. p ]Our proof incorporates the use of the noise operator with a random noise rate and some extremal properties of the Chebyshev polynomials.

Original language | American English |
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Pages (from-to) | 389-412 |

Number of pages | 24 |

Journal | Israel Journal of Mathematics |

Volume | 160 |

DOIs | |

State | Published - Aug 2007 |

Externally published | Yes |

### Bibliographical note

Funding Information:** This work was completed while the author was at the Institute for Advanced Study, Princeton, NJ. The material is based upon work supported in part by the National Science Foundation under agreement No. CCR-0324906. Recommen-dations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. Received July 21, 2005

Funding Information:

* Research supported in part by the Israel Science Foundation, grant no. 0329745.