Banach Journal of Mathematical Analysis

Linear dependency of translations and square-integrable representations

Peter A. Linnell, Michael J. Puls, and Ahmed Roman

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Let $G$ be a locally compact group. We examine the problem of determining when nonzero functions in $L^{2}(G)$ have linearly independent left translations. In particular, we establish some results for the case when $G$ has an irreducible, square-integrable, unitary representation. We apply these results to the special cases of the affine group, the shearlet group, and the Weyl–Heisenberg group. We also investigate the case when $G$ has an abelian, closed subgroup of finite index.

Article information

Banach J. Math. Anal. Volume 11, Number 4 (2017), 945-962.

Received: 28 January 2017
Accepted: 21 June 2017
First available in Project Euclid: 8 September 2017

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Digital Object Identifier

Primary: 43A80: Analysis on other specific Lie groups [See also 22Exx] 42C99: None of the above, but in this section
Secondary: 43A65: Representations of groups, semigroups, etc. [See also 22A10, 22A20, 22Dxx, 22E45]

affine group Atiyah conjecture left translations linear independence shearlet group square-integrable representation Weyl–Heisenberg group


Linnell, Peter A.; Puls, Michael J.; Roman, Ahmed. Linear dependency of translations and square-integrable representations. Banach J. Math. Anal. 11 (2017), no. 4, 945--962. doi:10.1215/17358787-2017-0028.

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