The aim of this paper is to study some properties of left translates of a square integrable function on the Heisenberg group. First, a necessary and sufficient condition for the existence of the canonical dual to a function \(\varphi \in L^{2}(\mathbb {R}^{2n})\) is obtained in the case of twisted shift-invariant spaces. Further, characterizations of \(\ell ^{2}\)-linear independence and the Hilbertian property of the twisted translates of a function \(\varphi \in L^{2}(\mathbb {R}^{2n})\) are obtained. Later these results are shown in the case of the Heisenberg group.

中文翻译：

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海森堡群上平方可积函数的左平移

本文的目的是研究海森堡群上平方可积函数的左平移的一些性质。首先，在扭曲移位的情况下，获得了对函数\（\ varphi \ in L ^ {2}（\ mathbb {R} ^ {2n}）\）的规范对偶的存在的充要条件-不变空间。此外，\（\ ell ^ {2} \）的线性独立性和函数\（\ varphi \ in L ^ {2}（\ mathbb {R} ^ {2n}）的扭曲平移的希尔伯特性质的表征\）。后来，这些结果在海森堡小组中得到了证明。