A simple reduction from a biased measure on the discrete cube to the uniform measure

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Abstract

We show that certain statements related to the Fourier-Walsh expansion of functions with respect to a biased measure on the discrete cube can be deduced from the respective results for the uniform measure by a simple reduction. In particular, we present simple generalizations to the biased measure μ p of the Bonami-Beckner hypercontractive inequality, and of Talagrand's lower bound on the size of the boundary of subsets of the discrete cube. Our generalizations are tight up to constant factors.

Original languageEnglish
Pages (from-to)1943-1957
Number of pages15
JournalEuropean Journal of Combinatorics
Volume33
Issue number8
DOIs
StatePublished - Nov 2012

Bibliographical note

Funding Information:
The author was partially supported by the Adams Fellowship Program of the Israeli Academy of Sciences and Humanities and by the Koshland Center for Basic Research .

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