## Banach Journal of Mathematical Analysis

### Linear dependency of translations and square-integrable representations

#### Abstract

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

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

Dates
Accepted: 21 June 2017
First available in Project Euclid: 8 September 2017

https://projecteuclid.org/euclid.bjma/1504836178

Digital Object Identifier
doi:10.1215/17358787-2017-0028

#### Citation

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. https://projecteuclid.org/euclid.bjma/1504836178

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