A new convolution theorem associated with the linear canonical transform

TL;DR

This paper introduces a new convolution operator linked to the linear canonical transform (LCT), establishing its algebraic properties and deriving a generalized convolution theorem and Young's inequality, with applications to convolution equations.

Contribution

It presents a novel canonical convolution operator for the LCT, proving its properties and extending classical theorems in signal processing.

Findings

The new convolution operator is commutative, associative, and distributive.

Generalized convolution theorem and Young's inequality are valid for the new operator.

Conditions for solving convolution equations with the LCT are established.

Abstract

In this paper, we first introduce a new notion of canonical convolution operator, and show that it satisfies the commutative, associative, and distributive properties, which may be quite useful in signal processing. Moreover, it is proved that the generalized convolution theorem and generalized Young's inequality are also hold for the new canonical convolution operator associated with the LCT. Finally, we investigate the sufficient and necessary conditions for solving a class of convolution equations associated with the LCT.