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Best constants in Kahane-Khintchine inequalities for complex Steinhaus functions. (English) Zbl 0841.41024

Summary: Let \(\{\varphi_k \}_{k\geq 1}\) be a sequence of independent random variables uniformly distributed on \([0,2\pi [\), and let \(|\cdot |_\psi\) denote the Orlicz norm induced by the function \(\psi(x)= \exp(|x|^2) -1\). Then \[ \Biggl|\sum^n_{k=1} z_k e^{i\varphi_k} \Biggr|_\psi \leq\sqrt {2} \Biggl( \sum^n_{k=1} |z_k |^2 \Biggr)^{1/2} \] for all \(z_1, \dots, z_n\in \mathbb{C}\) 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.

MSC:

41A44 Best constants in approximation theory
41A50 Best approximation, Chebyshev systems
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
60E15 Inequalities; stochastic orderings
60G50 Sums of independent random variables; random walks
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