Tensor convolutions and Hankel tensors

Abstract

Let A be an mth order n-dimensional tensor, where m, n are some positive integers and N:= m(n−1). Then A is called a Hankel tensor associated with a vector v ∈ ℝN+1 if Aσ = v k for each k = 0, 1,...,N whenever σ = (i1,..., im) satisfies i1 +· · ·+im = m+k. We introduce the elementary Hankel tensors which are some special Hankel tensors, and present all the eigenvalues of the elementary Hankel tensors for k = 0, 1, 2. We also show that a convolution can be expressed as the product of some third-order elementary Hankel tensors, and a Hankel tensor can be decomposed as a convolution of two Vandermonde matrices following the definition of the convolution of tensors. Finally, we use the properties of the convolution to characterize Hankel tensors and (0,1) Hankel tensors.