Linear canonical transform (LCT), which is of importance in solving differential equations and analyzing optical systems, extends the conventional Fourier transform to the time-frequency domain characterized by a parameter matrix $$\mathbf{A }=(a,b;c,d)$$ A = ( a , b ; c , d ) . The main objective of this paper is to study the Balian–Low theorem in the LCT domain. We first state a Balian–Low theorem associated with the LCT, indicating that if a function’s Gabor system forms a Riesz basis for $$L^2({\mathbb {R}})$$ L 2 ( R ) then it must be poorly localized in either time domain or LCT domains satisfying $$\frac{a}{b}\in {\mathbb {Z}}$$ a b ∈ Z . We then show that the symmetrically weighted version derived can hold for arbitrary LCT parameters under an assumption that the function is real-valued differentiable. Namely, if a real-valued differentiable function’s Gabor system forms a Riesz basis for $$L^2({\mathbb {R}})$$ L 2 ( R ) , then the product of its spreads in time and any LCT domains must be infinite.

中文翻译：

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与线性正则变换相关的广义 Balian-Low 定理

线性规范变换 (LCT) 在求解微分方程和分析光学系统方面具有重要意义，它将传统的傅立叶变换扩展到由参数矩阵 $$\mathbf{A }=(a,b;c ,d)$$ A = ( a , b ; c , d ) 。本文的主要目的是研究 LCT 域中的 Balian-Low 定理。我们首先陈述与 LCT 相关的 Balian-Low 定理，表明如果一个函数的 Gabor 系统形成了 $$L^2({\mathbb {R}})$$ L 2 ( R ) 的 Riesz 基，那么它必须是在满足 $$\frac{a}{b}\in {\mathbb {Z}}$$ ab ∈ Z 的时域或 LCT 域中本地化不佳。然后我们表明，在函数是实值可微的假设下，导出的对称加权版本可以适用于任意 LCT 参数。即，