论文标题
小波空间中的插值和hrt-conture
Interpolation in Wavelet Spaces and the HRT-Conjecture
论文作者
论文摘要
我们研究了小波空间$ \ MATHCAL {w} _ {g}（\ Mathcal {h}_π）\ subset l^{2}（g）$源自Square Antigeable Antegrable表示$ g \ t to \ g \ to \ Mathcal {\ Mathcal {u}（\ Mathcal {\ Mathcal {\ Mathcal {h} _ $ g $ g $ g $ g $我们表明，从它们之间的非平凡相交则施加了强烈的条件，小波空间是刚性的。此外，我们将其用于导致与凸性和正类型功能相关的小波变换的后果。由小波空间的繁殖核希尔伯特空间结构的促进，我们检查了一个插值问题。在时间频分析的设置中，此问题相当于HRT键入。最后，我们考虑了是否所有小波空间$ \ MATHCAL {w} _ {g}（\ Mathcal {h}_π）$ g $的$ g $集体用尽环境空间$ l^{2}（g）$。我们表明，答案对于紧凑型群体是肯定的，而对海森堡群体的负面影响是负面的。
We investigate the wavelet spaces $\mathcal{W}_{g}(\mathcal{H}_π)\subset L^{2}(G)$ arising from square integrable representations $π:G \to \mathcal{U}(\mathcal{H}_π)$ 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 conditions. 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 $\mathcal{W}_{g}(\mathcal{H}_π)$ 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.
