Generalized Balian–Low Theorem Associated with the Linear Canonical Transform

Abstract

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 $$\mathbf{A }=(a,b;c,d)$$ . 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 $$L^2({\mathbb {R}})$$ then it must be poorly localized in either time domain or LCT domains satisfying $$\frac{a}{b}\in {\mathbb {Z}}$$ . 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 $$L^2({\mathbb {R}})$$ , then the product of its spreads in time and any LCT domains must be infinite.