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

2010/01/07 by Nathan Keller, Keller, Nathan
Mathematics · #05D40 #06E30 #60C05 #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Approximation and Integration #Probability (math.PR) #math.CO #math.PR #msc:05D40 #msc:06E30 #msc:60C05

paper · pdf · doi:10.48550/arxiv.1001.1167

18 pages

openalex publication_date 2010/01/07 · arxiv created 2010/11/24 · arxiv updated 2010/11/25 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28

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.

Citations

• Logarithmic Sobolev inequalities for finite Markov chains

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