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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Best constants in Kahane-Khintchine inequalities for complex Steinhaus functions


Author: Goran Peškir
Journal: Proc. Amer. Math. Soc. 123 (1995), 3101-3111
MSC: Primary 46E30; Secondary 41A50, 60E15, 60G50
MathSciNet review: 1283561
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Abstract: Let $ {\{ {\varphi _k}\} _{k \geq 1}}$ be a sequence of independent random variables uniformly distributed on $ [0,2\pi [$, and let $ {\left\Vert \bullet \right\Vert _\psi }$ denote the Orlicz norm induced by the function $ \psi (x) = \exp (\vert x{\vert^2}) - 1$. Then

$\displaystyle {\left\Vert {\sum\limits_{k = 1}^n {{z_k}{e^{i{\varphi _k}}}} } \... ...q \sqrt 2 {\left( {\sum\limits_{k = 1}^n {\vert{z_k}{\vert^2}} } \right)^{1/2}}$

for all $ {z_1}, \ldots ,{z_n} \in {\mathbf{C}}$ and all $ n \geq 1$. The constant $ \sqrt 2 $ is shown to be the best possible. The method of proof relies upon a combinatorial argument, Taylor expansion, and the central limit theorem. The result is additionally strengthened by showing that the underlying functions are Schur-concave. The proof of this fact uses a result on the multinomial distribution of Rinott, and Schur's proposition on the sum of convex functions. The estimates obtained throughout are shown to be the best possible. The result extends and generalizes to provide similar inequalities and estimates for other Orlicz norms.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1283561-0
PII: S 0002-9939(1995)1283561-0
Keywords: Kahane-Khintchine inequality, complex Steinhaus sequence, Bernoulli sequence, Schur-convex (-concave), the gauge norm, Orlicz norm, Gaussian distribution, multinomial distribution
Article copyright: © Copyright 1995 American Mathematical Society