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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.

Keywords:  compact Kac algebra, Fourier transform, character, Sidon set, strong Sidon set, Helgason-Sidon set

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