In recent years, the quaternion analysis accounts for the correlated nature of the function components, hence, quaternion algebra became a well established mathematical discipline, and an active area of research with numerous applications in both pure and applied mathematics, such as quaternionic matrix analysis, color image processing, random signal processing, partial differential equations theory and so on. The quaternion Fourier transforms(QFTs) and quaternion linear canonical transforms (QLCTs)are the key tools in studying quaternionic signal processing. However, many fundamental and useful theorems of QFTs and QLCTs are not yet developed, such as inversion theorems, convolution theorems, Bedrosiaon theorems and so on. The aims of the thesis are studied the theory and applications of QFTs and QLCTs in signal processing. In Chapter 3, the inversion problems of various types of QFT's and QLCT's are studied. Theoretical background of quaternion bounded variation functions are investigated. Then the quaternion Fourier and linear canonical inversion formulas are derived. On the other hand, the quaternion Fourier and linear canonical inversion theorems also work for the absolutely integrable space. In Chapter 4, two novel types of convolution operators for QLCT are proposed. The first type is defined on the special domain, while the second type is defined on the/QLCT frequency domain. Then various types of convolution formulas are considered. Specially, the QLCT of convolution of two quaternionic functions can be implemented by product of their QLCT, or the summation of products of their QLCTs. As applications, correlation operators of QLCT are proposed and the correlation theorems are derived. Moreover, Fredholm integral equation of the first kind involving special kernels can be solved by the convolution formula. Thirdly, some system of second order partial differential equations, which can be transformed into the second order quaternion partial differential equations, can be solved by the convolution formula. Finally, it is convenient to design the multiplicative filters in the QLCT domain by the convolution theorem. There are three goals in Chapter 5. First, we develop the Bedrosian identities associated with quaternion partial and total Hilbert transforms. Second, necessary and sufficient conditions of quaternion analytic signal are investigated. Third, sufficient conditions of analyticity for the products of quaternion analytic signals are derived. Meanwhile, sufficient conditions of holomorphic property for products of quaternion holomorphic functions are also studied. The theories under QFT and QLCT are considered accordantly. Finally, conclusions and potential research topics are drawn at the end of this thesis. er study.