On a Solution of the Multidimensional Truncated Matrix-Valued Moment Problem

IF 1.2 3区 数学 Q1 MATHEMATICS
David P. Kimsey, Matina Trachana
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引用次数: 6

Abstract

We will consider the multidimensional truncated \(p \times p\) Hermitian matrix-valued moment problem. We will prove a characterisation of truncated \(p \times p\) Hermitian matrix-valued multisequence with a minimal positive semidefinite matrix-valued representing measure via the existence of a flat extension, i.e., a rank preserving extension of a multivariate Hankel matrix (built from the given truncated matrix-valued multisequence). Moreover, the support of the representing measure can be computed via the intersecting zeros of the determinants of matrix-valued polynomials which describe the flat extension. We will also use a matricial generalisation of Tchakaloff’s theorem due to the first author together with the above result to prove a characterisation of truncated matrix-valued multisequences which have a representing measure. When \(p = 1\), our result recovers the celebrated flat extension theorem of Curto and Fialkow. The bivariate quadratic matrix-valued problem and the bivariate cubic matrix-valued problem are explored in detail.

多维截断矩阵值矩问题的一种解
我们将考虑多维截断\(p \times p\)厄米矩阵值矩问题。我们将通过一个平面扩展的存在性证明截断的\(p \times p\)荷米特矩阵值多序列的一个特征,即多元汉克尔矩阵的一个保秩扩展(从给定的截断的荷米特矩阵值多序列建立)。此外,表示测度的支持度可以通过描述平面扩展的矩阵值多项式的行列式的相交零来计算。我们还将利用第一作者对Tchakaloff定理的一个物质推广,结合上述结果,证明具有表示测度的截断矩阵值多序列的一个表征。当\(p = 1\)时,我们的结果恢复了Curto和Fialkow著名的平面扩展定理。详细探讨了二元二次矩阵值问题和二元三次矩阵值问题。
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来源期刊
CiteScore
2.60
自引率
0.00%
发文量
23
审稿时长
>12 weeks
期刊介绍: Milan Journal of Mathematics (MJM) publishes high quality articles from all areas of Mathematics and the Mathematical Sciences. The authors are invited to submit "articles with background", presenting a problem of current research with its history and its developments, the current state and possible future directions. The presentation should render the article of interest to a wider audience than just specialists. Many of the articles will be "invited contributions" from speakers in the "Seminario Matematico e Fisico di Milano". However, also other authors are welcome to submit articles which are in line with the "Aims and Scope" of the journal.
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