On the solvability of the indefinite Hamburger moment problem

Yongjian Hu; Huifeng Hao; Xuzhou Zhan; Yongjian Hu; Huifeng Hao; Xuzhou Zhan

doi:10.3934/math.20231535

AIMS Mathematics

2023, Volume 8, Issue 12: 30023-30037. doi: 10.3934/math.20231535

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On the solvability of the indefinite Hamburger moment problem

1.
School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
2.
Department of Statistics, Hebei University of Engineering, Handan 056038, China
3.
Department of Mathematics, Beijing Normal University at Zhuhai, Zhuhai 519087, China

Received: 14 July 2023 Revised: 21 October 2023 Accepted: 31 October 2023 Published: 06 November 2023
MSC : 30E05, 15B57, 46C20

In this paper, we present a new approach for the solvability of the indefinite Hamburger moment problem in the class of generalized Nevanlinna functions with a given negative index, which is more algebraic and completely different from the existing method ^[8] based on the step-by-step Schur algorithm. As a by-product of this approach, we simultaneously obtain a concrete rational solution of such an indefinite Hamburger moment problem when the solvability conditions are met. The basic strategy focuses on the structural characteristics of the Hankel matrix and the relation among the Hankel, Loewner, Bezout and some other structured matrices.
- generalized Nevanlinna function,
- infinite Hamburger moment problem,
- Hankel matrix,
- characteristic degree,
- characteristic polynomial quadruple,
- quasidirect decomposition,
- McMillan degree
Citation: Yongjian Hu, Huifeng Hao, Xuzhou Zhan. On the solvability of the indefinite Hamburger moment problem[J]. AIMS Mathematics, 2023, 8(12): 30023-30037. doi: 10.3934/math.20231535

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Abstract

In this paper, we present a new approach for the solvability of the indefinite Hamburger moment problem in the class of generalized Nevanlinna functions with a given negative index, which is more algebraic and completely different from the existing method ^[8] based on the step-by-step Schur algorithm. As a by-product of this approach, we simultaneously obtain a concrete rational solution of such an indefinite Hamburger moment problem when the solvability conditions are met. The basic strategy focuses on the structural characteristics of the Hankel matrix and the relation among the Hankel, Loewner, Bezout and some other structured matrices.

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