Journal of Operator Theory

Volume 70, Issue 2, Autumn 2013 pp. 375-399.

$L^1$ -Norm estimates of character sums defined by a Sidon set in the dual of a compact Kac algebra

Authors: Tobias Blendek (2) and Johannes Michalicek (2)
Author institution: (1) Department of Mathematics and Statistics, Helmut Schmidt University Hamburg, Holstenhofweg 85, 22043 Hamburg, Germany
(2) Department of Mathematics, University of Hamburg, Bundesstra{\ss}e 55, 20146 Hamburg, Germany

Summary: We generalize the following fact to compact Kac algebras: Let

$G$ be a compact abelian group, and let

$f$ be any trigonometric polynomial on

$G$ , whose Fourier transform

$\widehat{f}$ vanishes outside of a Sidon set

$E$ in the dual, discrete abelian group

$\Gamma$ of

$G$ . Then we have

$\|f\|_2\leqslant K_E\|f\|_1$ , where

$K_E$ is a constant depending only on

$E$ . For this generalization, we introduce the notion of Helgason--Sidon sets, which is based on S.~Helgason's work on lacunary Fourier series on arbitrary compact groups. We establish the above inequality for all finite linear combinations of characters defined by a Helgason--Sidon set in the set of all minimal central projections.

DOI: http://dx.doi.org/10.7900/jot.2011sep03.1945
Keywords: compact Kac algebra, Fourier transform, character, Sidon set, strong Sidon set, Helgason-Sidon set

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