  Novel types of convolution operators for quaternion linear canonical
transform (QLCT) are proposed. Type one and two are defined in the spatial and
QLCT spectral domains, respectively. They are distinct in the quaternion space
and are consistent once in complex or real space. Various types of convolution
formulas are discussed. Consequently, the QLCT of the convolution of two
quaternionic functions can be implemented by the product of their QLCTs, or the
summation of the products of their QLCTs. As applications, correlation
operators and theorems of the QLCT are derived. The proposed convolution
formulas are used to solve Fredholm integral equations with special kernels.
Some systems of second-order partial differential equations, which can be
transformed into the second-order quaternion partial differential equations,
can be solved by the convolution formulas as well. As a final point, we
demonstrate that the convolution theorem facilitates the design of
multiplicative filters.
