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

Contents Full-Text PDF