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Charges solve the truncated complex moment problem

Center of Mathematical Research of Rabat, P. O. Box 1014, Department of Mathematics, Faculty of Sciences, Mohammed V University, Rabat, Morocco

E-mail Address: kaissar.idrissi@gmail.com

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and

E. H. Zerouali

Center of Mathematical Research of Rabat, P. O. Box 1014, Department of Mathematics, Faculty of Sciences, Mohammed V University, Rabat, Morocco

E-mail Address: Zerouli@fsr.ac.ma

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https://doi.org/10.1142/S0219025718500273Cited by:1 (Source: Crossref)

Abstract

Let $γ \equiv {γ_{i j}}_{(i, j) \in I}$ , with $I \subseteq ℤ_{+} \times ℤ_{+}$ and $\bar{γ_{i j}} = γ_{j i}$ , be a given complex-valued sequence. The complex moment problem (respectively, the general complex moment problem) associated with $γ$ consists in determining necessary and sufficient conditions for the existence of a positive Borel measure (respectively, a charge) $μ$ on $ℂ$ such that

γ i j = \int ℂ ¯ z i z j d μ, for (i, j) \in I . <math display="block" altimg="eq-00007.gif"><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo class="MathClass-rel">=</mo><msub><mrow><mo class="MathClass-op">\int</mo></mrow><mrow><mi>ℂ</mi></mrow></msub><msup><mrow><mover accent="false" class="mml-overline"><mrow><mi>z</mi></mrow><mo accent="true">¯</mo></mover></mrow><mrow><mi>i</mi></mrow></msup><msup><mrow><mi>z</mi></mrow><mrow><mi>j</mi></mrow></msup><mi>d</mi><mi>μ</mi><mo class="MathClass-punc">,</mo><mspace width="1em" class="quad"></mspace><mstyle><mtext>for </mtext></mstyle><mo class="MathClass-open" stretchy="false">(</mo><mi>i</mi><mo class="MathClass-punc">,</mo><mi>j</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-rel">\in</mo><mi>I</mi><mo class="MathClass-punc">.</mo></mrow></math>

In this paper, we investigate the notion of recursiveness in the two variable case. We obtain several useful results that we use to deduce new necessary and sufficient conditions for the truncated complex moment problem to admit a solution. In particular, we show that the general complex moment problem always has a solution. A concrete construction of the solution and an illustrating example are also given.

Communicated by Luigi Accardi

Keywords:

AMSC: 47A57, 30E05, 44A60, 42A82

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