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The Truncated Hamburger Moment Problems with Gaps in the Index Set

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Abstract

In this article we solve four special cases of the truncated Hamburger moment problem (THMP) of degree 2k with one or two missing moments in the sequence. As corollaries we obtain via appropriate substitutions, the solutions to bivariate truncated moment problems of degree 2k for the curves \(y=x^3\) (first solved by Fialkow in Trans Am Math Soc 363:3133–3165, 2011), \(y^2=x^3\), \(y=x^4\) where a certain moment of degree \(2k+1\) is known and \(y^3=x^4\) with a certain moment given. The main technique is the completion of the partial positive semidefinite matrix (ppsd) such that the conditions of Curto and Fialkow’s solution of the THMP are satisfied. The main tools are the use of the properties of positive semidefinite Hankel matrices and a result on all completions of a ppsd matrix with one unknown entry, proved by the use of the Schur complements for \(2\times 2\) and \(3\times 3\) block matrices.

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Acknowledgements

I would like to thank Jaka Cimprič and Abhishek Bhardwaj for useful suggestions on the preliminary versions of this article.

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  1. Faculty of Computer and Information Science, University of Ljubljana, Večna pot 113, 1000, Ljubljana, Slovenia

    Aljaž Zalar

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  1. Aljaž Zalar
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Zalar, A. The Truncated Hamburger Moment Problems with Gaps in the Index Set. Integr. Equ. Oper. Theory 93, 22 (2021). https://doi.org/10.1007/s00020-021-02628-6

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