Hypercontractivity of Spherical Averages in Hamming Space
Name
15m1046575.pdf
Description
Published version
Size
461.39 KB
Format
Adobe PDF
Checksum (MD5)
936dd915b03a429d25c6c46b33fce732
Author(s)
Polyanskiy, Yury
Date Issued
2019
Journal
SIAM Journal on Discrete Mathematics
Publisher
Society for Industrial & Applied Mathematics (SIAM)
Version
Final published version
Abstract
© 2019 Society for Industrial and Applied Mathematics Consider the linear space of functions on the binary hypercube and the linear operator S\delta acting by averaging a function over a Hamming sphere of radius \delta n around every point. It is shown that this operator has a dimension-independent bound on the norm Lp \rightarrow L2 with p = 1 + (1 - 2\delta )2. This result evidently parallels a classical estimate of Bonami and Gross for Lp \rightarrow Lq norms for the operator of convolution with a Bernoulli noise. The estimate for S\delta is harder to obtain since the latter is neither a part of a semigroup nor a tensor power. The result is shown by a detailed study of the eigenvalues of S\delta and Lp \rightarrow L2 norms of the Fourier multiplier operators \Pi a with symbol equal to a characteristic function of the Hamming sphere of radius a (in the notation common in boolean analysis \Pi af = f=a, where f=a is a degree-a component of function f). A sample application of the result is given: Any set A \subset \BbbFn2 with the property that A + A contains a large portion of some Hamming sphere (counted with multiplicity) must have cardinality a constant multiple of 2n
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Laboratory for Information and Decision Systems
Massachusetts Institute of Technology. Institute for Data, Systems, and Society
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DOI of Published Version
https://doi.org/10.1137/15M1046575
