Commutativity conditions for the natural quaternion convolution in 1D

Determine non-trivial necessary and sufficient conditions on quaternion-valued functions f and g on the real line such that the natural quaternion convolution ostar, defined by Akila and Roopkumar for one-dimensional quaternion Fourier analysis, is commutative; specifically, characterize when f ostar g = g ostar f holds beyond the known trivial cases (both functions complex-valued or one function real-valued and even).

Background

In the one-dimensional quaternion Fourier transform setting, Akila and Roopkumar introduced a natural convolution operation (denoted ostar) tailored to quaternion-valued signals and proved a convolution theorem for it.

Due to the non-commutativity of quaternion multiplication, this convolution is generally non-commutative. The authors note that commutativity can be guaranteed in simple cases, such as when both functions are complex-valued or when one of the functions is a real-valued even function.

The survey explicitly highlights the unresolved task of identifying a precise and comprehensive characterization—necessary and sufficient conditions—under which the ostar convolution becomes commutative, thereby advancing the algebraic understanding of quaternion signal processing.