时频表示的伞形定理
Umbrella theorems for time-frequency representations
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中文总结 AI 辅助
研究\(L^2(\mathbb{R}^d)\)中与时频表示相关的伞形定理，借助梅林变换将分析扩展到\(L^2(\mathbb{R}^+)\)，给出相关结果，拓展了该定理在不同空间的研究。
中文摘要 AI 辅助
H.S. Shapiro提出的不确定性原理，即所谓的伞形定理，断言在\(L^2(\mathbb{R})\)中不存在平方可积函数能一致支配一个正交族的所有元素及其傅里叶变换，除非该序列是有限的。本文给出了一些关于\(L^2(\mathbb{R}^d)\)中与时频表示相关的伞形定理的结果。通过梅林变换将分析进一步扩展到\(L^2(\mathbb{R}^+)\)的情况。
英文摘要
An uncertainty principle due to H.S. Shapiro, the so-called Umbrella Theorem, asserts that there is no square integrable function uniformly dominating all the elements of an orthonormal family in $L^2(\mathbb{R})$ and their Fourier transforms, unless the sequence is finite. In this paper we present some results on Umbrella Theorems in $L^2(\mathbb{R}^d)$ related to time-frequency representations. We further extend the analysis to the case of $L^2(\mathbb{R}^+)$, by means of the Mellin transform.
