TY - JOUR
T1 - On the ℓ4:ℓ2 ratio of functions with restricted Fourier support
AU - Kirshner, Naomi
AU - Samorodnitsky, Alex
N1 - Publisher Copyright:
© 2020 Elsevier Inc.
PY - 2020/5
Y1 - 2020/5
N2 - Given a subset A⊆{0,1}n, let μ(A) be the maximal ratio between ℓ4 and ℓ2 norms of a function whose Fourier support is a subset of A.1 We make some simple observations about the connections between μ(A) and the additive properties of A on one hand, and between μ(A) and the uncertainty principle for A on the other hand. One application obtained by combining these observations with results in additive number theory is a stability result for the uncertainty principle on the discrete cube. Our more technical contribution is determining μ(A) rather precisely, when A is a Hamming sphere S(n,k) for all 0≤k≤n.
AB - Given a subset A⊆{0,1}n, let μ(A) be the maximal ratio between ℓ4 and ℓ2 norms of a function whose Fourier support is a subset of A.1 We make some simple observations about the connections between μ(A) and the additive properties of A on one hand, and between μ(A) and the uncertainty principle for A on the other hand. One application obtained by combining these observations with results in additive number theory is a stability result for the uncertainty principle on the discrete cube. Our more technical contribution is determining μ(A) rather precisely, when A is a Hamming sphere S(n,k) for all 0≤k≤n.
KW - Additive combinatorics
KW - Fourier spectrum
KW - Hypercontractivity
KW - Krawchouk polynomials
KW - Uncertainty principle
UR - http://www.scopus.com/inward/record.url?scp=85077735278&partnerID=8YFLogxK
U2 - 10.1016/j.jcta.2019.105202
DO - 10.1016/j.jcta.2019.105202
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AN - SCOPUS:85077735278
SN - 0097-3165
VL - 172
JO - Journal of Combinatorial Theory. Series A
JF - Journal of Combinatorial Theory. Series A
M1 - 105202
ER -