We investigate the wavelet spaces \(\mathcal {W}_{g}(\mathcal {H}_{\pi })\subset L^{2}(G)\) arising from square integrable representations \(\pi :G \rightarrow \mathcal {U}(\mathcal {H}_{\pi })\) of a locally compact group G. We show that the wavelet spaces are rigid in the sense that non-trivial intersection between them imposes strong restrictions. Moreover, we use this to derive consequences for wavelet transforms related to convexity and functions of positive type. Motivated by the reproducing kernel Hilbert space structure of wavelet spaces we examine an interpolation problem. In the setting of time–frequency analysis, this problem turns out to be equivalent to the HRT-conjecture. Finally, we consider the problem of whether all the wavelet spaces \(\mathcal {W}_{g}(\mathcal {H}_{\pi })\) of a locally compact group G collectively exhaust the ambient space \(L^{2}(G)\). We show that the answer is affirmative for compact groups, while negative for the reduced Heisenberg group.

我们研究由平方可积表示\ {\ pi产生的小波空间\ {\ mathcal {W} _ {g}（\ mathcal {H} _ {\ pi}} \ subset L ^ {2}（G）\）局部紧致群G的G \ rightarrow \ mathcal {U}（\ mathcal {H} _ {\ pi}）\）。我们表明小波空间是刚性的，因为它们之间的非平凡相交强加了约束。此外，我们用它来推导与正型凸性和函数有关的小波变换的结果。基于小波空间的再生希尔伯特空间结构，我们研究了插值问题。在时频分析的设置中，这个问题等同于HRT猜想。最后，我们考虑是否所有小波空间局部紧凑群G的\（\ mathcal {W} _ {g}（\ mathcal {H} _ {\ pi}} \）共同耗尽了周围空间\（L ^ {2}（G）\）。我们表明，对于紧凑型群体，答案是肯定的；对于减少型海森堡组，答案是肯定的。