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This article is cited in 4 scientific papers (total in 4 papers)
Khintchine Inequality for Sets of Small Measure
S. V. Astashkin Samara State University
Abstract:
The following theorem is proved. Let $r_i$ be the Rademacher functions, i.e., $r_i(t):=\operatorname{sign}\sin(2^i\pi t)$, $t\in[0,1]$, $i\in\mathbb{N}$. If a set $E\subset [0,1]$ satisfies the condition $m(E\cap (a,b))>0$ for any interval $(a,b)\subset [0,1]$, then, for some constant $\gamma=\gamma(E)>0$ depending only on $E$
and for all sequences $a=(a_k)_{k=1}^\infty\in\ell^2$,
$$
\int_E\bigg|\sum_{i=1}^\infty a_ir_i(t)\bigg|\,dt\ge \gamma \bigg(\sum_{i=1}^\infty a_i^2\bigg)^{1/2}.
$$
As a consequence of this result, a version of the weighted Khintchine inequality is obtained.
Keywords:
Rademacher functions, Khintchine inequality, $L_p$-spaces, Paley–Zygmund inequality.
Received: 04.03.2013
Citation:
S. V. Astashkin, “Khintchine Inequality for Sets of Small Measure”, Funktsional. Anal. i Prilozhen., 48:4 (2014), 1–8; Funct. Anal. Appl., 48:4 (2014), 235–241
Linking options:
https://www.mathnet.ru/eng/faa3169https://doi.org/10.4213/faa3169 https://www.mathnet.ru/eng/faa/v48/i4/p1
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Abstract page: | 463 | Full-text PDF : | 194 | References: | 81 | First page: | 30 |
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