Least squares Hermitian problem of a kind of quaternion tensor equation

Yuan and Liao [S.F. Yuan, A.P. Liao, Least squares Hermitian solution of the complex matrix equation AXB+CXD=E$$ AXB+ CXD=E $$ with the least norm. J. Frankl. Inst. 351 (2014), 4978-4997] gave a new product for complex matrices and vectors and obtained the least squares Hermitian solution with the least norm of complex matrix equation AXB+CXD=E$$ AXB+ CXD=E $$. In this paper, we generalize the product of complex matrices and vectors to quaternion tensors and vectors and consider the least squares Hermitian problem of quaternion tensor equation A*NX=B$$ \mathcal{A}{\ast}_N\mathcal{X}=\mathcal{B} $$, where the symbol *N$$ {\ast}_N $$ is the Einstein product of two tensors. Our main work is to derive the explicit expression of least squares Hermitian solution with the least norm and numerical algorithm to obtain the solution. We also provide numerical examples to justify the feasibility of our algorithm.