Abstract
The following theorem is proved. Let r i be the Rademacher functions, i.e., r i (t):= sign sin(2i πt), t ∈ [0, 1], i ∈ ℕ. If a set E ⊂ [0, 1] satisfies the condition m(E ∩ (a, b)) > 0 for any interval (a, b) ⊂ [0, 1], then, for some constant γ = γ(E) > 0 depending only on E and for all sequences \(a = (a_k )_{k = 1}^\infty \in \ell ^2 \),
. As a consequence of this result, a version of the weighted Khintchine inequality is obtained.
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Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 48, No. 4, pp. 1–8, 2014
Original Russian Text Copyright © by S. V. Astashkin
This research was partially supported by RFBR grant no. 12-01-00198.
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Astashkin, S.V. Khintchine inequality for sets of small measure. Funct Anal Its Appl 48, 235–241 (2014). https://doi.org/10.1007/s10688-014-0066-8
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DOI: https://doi.org/10.1007/s10688-014-0066-8