In recent years, the Linear canonical transform (LCT) has been recognized as a powerful tool in signal processing scenarios and optics. In this two-part paper, the issue of unitary and Hermitian operators and their duality concept associated with LCT are addressed. In Part I, based on the proposed operators, three LCT-related topics are derived. Firstly, the definitions of convolution and correlation operations in LCT domain are constructed. Secondly, a new transform which is named as Linear canonical mellin transform (LCMT) is introduced and then convolution and correlation operations in LCMT domain are given. Thirdly, the so-called joint canonical distributions which satisfy the canonical transform marginal are given. Part II concerns two applications of the theory derived in Part I for signal analysis.