The linear canonical transform plays an important role in engineering and many applied fields, as it is the case of optics and signal processing. In this paper, a new convolution for the linear canonical transform is proposed and a corresponding product theorem is deduced. It is also proved a generalized Young’s inequality for the introduced convolution operator. Moreover, necessary and sufficient conditions are obtained for the solvability of a class of convolution type integral equations associated with the linear canonical transform. Finally, the obtained results are implemented in multiplicative filters design, through the product in both the linear canonical transform domain and the time domain, where specific computations and comparisons are exposed.

线性规范变换在光学和信号处理的情况下，在工程和许多应用领域中都起着重要作用。本文提出了一种新的线性正则变换卷积，并推导了相应的乘积定理。也证明了所引入的卷积算子的广义杨氏不等式。此外，获得了与线性典范变换相关的一类卷积型积分方程的可解性的必要和充分条件。最后，通过乘积在线性规范变换域和时域中的乘积，在乘法滤波器设计中实现了结果，其中公开了特定的计算和比较。