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In this paper, we show that if a lower-order Hankel tensor is positivesemi-definite (or positive definite, or negative semi-definite, or negativedefinite, or SOS), then its associated higher-order Hankel tensor with the samegenerating vector, where the higher order is a multiple of the lower order, isalso positive semi-definite (or positive definite, or negative semi-definite,or negative definite, or SOS, respectively). Furthermore, in this case, theextremal H-eigenvalues of the higher order tensor are bounded by the extremalH-eigenvalues of the lower order tensor, multiplied with some constants. Basedon this inheritance property, we give a concrete sum-of-squares decompositionfor each strong Hankel tensor. Then we prove the second inheritance property ofHankel tensors, i.e., a Hankel tensor has no negative (or non-positive, orpositive, or nonnegative) H-eigenvalues if the associated Hankel matrix of thatHankel tensor has no negative (or non-positive, or positive, or nonnegative,respectively) eigenvalues. In this case, the extremal H-eigenvalues of theHankel tensor are also bounded by the extremal eigenvalues of the associatedHankel matrix, multiplied with some constants. The third inheritance propertyof Hankel tensors is raised as a conjecture.

Inheritance Properties and Sum-of-Squares Decomposition of Hankel Tensors: Theory and Algorithms