Glasnik Matematicki, Vol. 50, No. 1 (2015), 193-205.

ON QUASI-GREEDY BASES ASSOCIATED WITH UNITARY REPRESENTATIONS OF COUNTABLE GROUPS

Morten Nielsen

Department of Mathematical Sciences, Aalborg University , DK-9220 Aalborg, Denmark
e-mail: mnielsen@math.aau.dk


Abstract.   We consider the natural generating system for a cyclic subspace of a Hilbert space generated by a dual integrable unitary representation of a countable abelian group. We prove, under mild hypothesis, that whenever the generating system is a quasi-greedy basis it must also be an unconditional Riesz basis. A number of applications to Gabor systems and to general Vilenkin systems are considered. In particular, we show that any Gabor Schauder basis that also forms a quasi-greedy system in L2 is in fact a Riesz basis, and therefore satisfies the classical Balian-Low theorem.

2010 Mathematics Subject Classification.   42C10, 41A45.

Key words and phrases.   Quasi-greedy bases, dual integrable representation, Gabor systems, integer translates, Vilenkin system.


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DOI: 10.3336/gm.50.1.11


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