# On the best constant in the Khinchin-Kahane inequality

Studia Mathematica (1994)

• Volume: 109, Issue: 1, page 101-104
• ISSN: 0039-3223

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## Abstract

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We prove that if ${r}_{i}$ is the Rademacher system of functions then ${\left(ʃ\parallel {\sum }_{i=1}^{n}{x}_{i}{r}_{i}\left(t\right){\parallel }^{2}dt\right)}^{1/2}\le \surd 2ʃ\parallel {\sum }_{i=1}^{n}{x}_{i}{r}_{i}\left(t\right)\parallel dt$ for any sequence of vectors ${x}_{i}$ in any normed linear space F.

## How to cite

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Latała, Rafał, and Oleszkiewicz, Krzysztof. "On the best constant in the Khinchin-Kahane inequality." Studia Mathematica 109.1 (1994): 101-104. <http://eudml.org/doc/216056>.

@article{Latała1994,
abstract = {We prove that if $r_i$ is the Rademacher system of functions then $(ʃ ∥∑_\{i=1\}^\{n\} x_\{i\}r_\{i\}(t)∥^2 dt)^\{1/2\} ≤ √2 ʃ ∥∑_\{i=1\}^\{n\}x_\{i\}r_\{i\}(t)∥dt$ for any sequence of vectors $x_i$ in any normed linear space F.},
author = {Latała, Rafał, Oleszkiewicz, Krzysztof},
journal = {Studia Mathematica},
keywords = {vector-valued inequalities; Rademacher system of functions},
language = {eng},
number = {1},
pages = {101-104},
title = {On the best constant in the Khinchin-Kahane inequality},
url = {http://eudml.org/doc/216056},
volume = {109},
year = {1994},
}

TY - JOUR
AU - Latała, Rafał
AU - Oleszkiewicz, Krzysztof
TI - On the best constant in the Khinchin-Kahane inequality
JO - Studia Mathematica
PY - 1994
VL - 109
IS - 1
SP - 101
EP - 104
AB - We prove that if $r_i$ is the Rademacher system of functions then $(ʃ ∥∑_{i=1}^{n} x_{i}r_{i}(t)∥^2 dt)^{1/2} ≤ √2 ʃ ∥∑_{i=1}^{n}x_{i}r_{i}(t)∥dt$ for any sequence of vectors $x_i$ in any normed linear space F.
LA - eng
KW - vector-valued inequalities; Rademacher system of functions
UR - http://eudml.org/doc/216056
ER -

## References

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1. [1] U. Haagerup, The best constants in the Khintchine inequality, Studia Math. 70 (1982), 231-283. Zbl0501.46015
2. [2] J.-P. Kahane, Sur les sommes vectorielles $\sum ±{u}_{n}$, C. R. Acad. Sci. Paris 259 (1964), 2577-2580.
3. [3] A. Khintchine [A. Khinchin], Über dyadische Brüche, Math. Z. 18 (1923), 109-116.
4. [4] S. J. Szarek, On the best constants in the Khinchin inequality, Studia Math. 58 (1976), 197-208. Zbl0424.42014
5. [5] B. Tomaszewski, Two remarks on the Khintchin-Kahane inequality, Colloq. Math. 46 (1982), 283-288. Zbl0501.46021
6. [6] B. Tomaszewski, A simple and elementary proof of the Khintchine inequality with the best constant, Bull. Sci. Math. (2) 111 (1987), 103-109. Zbl0623.42015

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