Minimax Theory and its Applications 02 (2017), No. 2, 231--248
Copyright Heldermann Verlag 2017
New Classes of Positive Semi-Definite Hankel Tensors
Qun Wang
Dept. of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
wangqun876@gmail.com
Guoyin Li
Dept. of Applied Mathematics, University of New South Wales, Sydney 2052, Australia
g.li@unsw.edu.au
Liqun Qi
Dept. of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
maqilq@polyu.edu.hk
Yi Xu
Dept. of Mathematics, Southeast University, Nanjing 210096, P. R. China
yi.xu1983@gmail.com
A Hankel tensor is called a strong Hankel tensor if the Hankel matrix generated by its generating vector is positive semi-definite. It is known that an even order strong Hankel tensor is a sum-of-squares tensor, and thus a positive semi-definite tensor. The SOS decomposition of strong Hankel tensors has been well-studied by W. Ding, L. Qi and Y. Wei [Inheritance properties and sum-of-squares decomposition of Hankel tensors: theory and algorithms, BIT Numerical Mathematics (2016)]. On the other hand, very little is known for positive semi-definite Hankel tensors which are not strong Hankel tensors. In this paper, we study some classes of positive semi-definite Hankel tensors which are not strong Hankel tensors. These include truncated Hankel tensors and quasi-truncated Hankel tensors. Then we show that a strong Hankel tensor generated by an absoluate integrable function is always completely decomposable, and give a class of SOS Hankel tensors which are not completely decomposable.
Keywords: Hankel tensors, generating vectors, positive semi-definiteness, strong Hankel tensors.
MSC: 15A18, 15A69.
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