Hypercontractive Inequalities for the Second Norm of Highly Concentrated Functions,and Mrs. Gerber’s-Type Inequalities for the Second Rényi Entropy期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

Hypercontractive Inequalities for the Second Norm of Highly Concentrated Functions,and Mrs. Gerber’s-Type Inequalities for the Second Rényi Entropy

Authors:	Niv Levhari Alex Samorodnitsky

Affiliation:	1.School of Engineering and Computer Science, The Hebrew University of Jerusalem, Jerusalem 9103401, Israel;2.School of Mathematical Sciences, Tel Aviv University, Tel Aviv 6997801, Israel

Abstract:	Let $T_{ϵ}$ , $0 \leq ϵ \leq 1 / 2$ , be the noise operator acting on functions on the boolean cube ${0, 1}^{n}$ . Let f be a distribution on ${0, 1}^{n}$ and let $q > 1$ . We prove tight Mrs. Gerber-type results for the second Rényi entropy of $T_{ϵ} f$ which take into account the value of the $q^{t h}$ Rényi entropy of f. For a general function f on ${0, 1}^{n}$ we prove tight hypercontractive inequalities for the $ℓ_{2}$ norm of $T_{ϵ} f$ which take into account the ratio between $ℓ_{q}$ and $ℓ_{1}$ norms of f.

Keywords:	entropy, hypercontractivity, Ré nyi entropy, Mrs. Gerber’ s inequality