Abstract
Let Tϵ, 0≤ϵ≤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 Tϵf which take into account the value of the qth Rényi entropy of f. For a general function f on {0,1}n we prove tight hypercontractive inequalities for the ℓ2 norm of Tϵf which take into account the ratio between ℓq and ℓ1 norms of f.
Original language | English |
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Article number | 1376 |
Journal | Entropy |
Volume | 24 |
Issue number | 10 |
DOIs | |
State | Published - 27 Sep 2022 |
Bibliographical note
Publisher Copyright:© 2022 by the authors.
Keywords
- Mrs. Gerber’s inequality
- Rényi entropy
- entropy
- hypercontractivity