Lp$$ {L}^p $$‐theory of linear canonical transforms and related uncertainty principles

Abstract

In this paper, we revisit some fundamental properties of linear canonical transform (abbreviated as LCT). In particular, we prove the additive property rigorously for LCT in the higher dimensional case (abbreviated as MLCT). We also consider the ‐theory of MLCT with . Specifically, the inversion theorem of MLCT by the related Gauss and Abel means is studied, and the pointwise convergence of approximate identities with respect to convolution for MLCT is also obtained. As applications, we study the ‐type Heisenberg‐Pauli‐Weyl uncertainty principles and the ‐type Donoho‐Stark uncertainty principles for MLCT.