Positive Semi-Definiteness of Generalized Anti-Circulant Tensors

Anti-circulant tensors have applications in exponential data fitting. They are special Hankel tensors. In this paper, we extend the definition of anti-circulant tensors to generalized anti-circulant tensors by introducing a circulant index $r$ such that the entries of the generating vector of a Hankel tensor are circulant with module $r$. In the special case when $r =n$, where $n$ is the dimension of the Hankel tensor, the generalized anticirculant tensor reduces to the anti-circulant tensor. Hence, generalized anti-circulant tensors are still special Hankel tensors. For the cases that $GCD(m, r) =1$, $GCD(m, r) = 2$ and some other cases, including the matrix case that $m=2$, we give necessary and sufficient conditions for positive semi-definiteness of even order generalized anti-circulant tensors, and show that in these cases, they are SOS tensors. This shows that, in these cases, there are no PNS (positive semidefinite tensors which are not sum of squares) Hankel tensors.