AI 中文总结
该研究建立了海森伯格群上短时傅里叶变换的不确定性原理系统理论，确立了多个经典不确定性原理的非交换类似物，扩展了相关现象的适用范围，凸显了非交换调和分析的作用。
AI 中文摘要
我们建立了海森伯格群上短时傅里叶变换（STFT）的不确定性原理的系统理论。基于Fischer等人以及后续Biswas-Thangavelu提出的海森伯格群上调制空间与时频分析的最新进展，我们确立了时频分析中若干基础不确定性原理的非交换类似物，包括Benedicks定理、Donoho-Stark不确定性原理和Lieb不等式。作为Lieb不等式的推论，我们推导了Hirschman型的基于熵的不确定性原理。我们进一步建立了海森伯格型不确定性不等式和Price风格的局部不确定性原理。此外，我们证明了Beurling-Hardy型定理，该定理刻画了衰减与相空间局域化之间的相互作用。最后，我们研究了STFT的衰减性质及其对时频集中的意义。这些结果将大量经典不确定性现象从欧几里得情形扩展到海森伯格群，凸显了非交换调和分析在相空间局域化研究中的作用。
英文摘要
We develop a systematic theory of uncertainty principles for the short-time Fourier transform (STFT) on the Heisenberg group. Building on recent developments in modulation spaces and time-frequency analysis on the Heisenberg group introduced by Fischer et al. and later by Biswas-Thangavelu, we establish noncommutative analogues of several fundamental uncertainty principles in time-frequency analysis, including Benedicks' theorem, the Donoho-Stark uncertainty principle, and Lieb's inequality. As a consequence of Lieb's inequality, we derive an entropy-based uncertainty principle of Hirschman type. We further establish a Heisenberg-type uncertainty inequality and local uncertainty principles in the spirit of Price. In addition, we prove a Beurling-Hardy-type theorem that captures the interplay between decay and phase-space localization. Finally, we investigate decay properties of the STFT and their implications for time-frequency concentration. These results extend a broad spectrum of classical uncertainty phenomena from the Euclidean setting to the Heisenberg group, highlighting the role of noncommutative harmonic analysis in the study of phase-space localization.
