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Stability of Third Degree Linear Functionals and Rational Spectral Transformations

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Abstract

In this paper, we consider linear forms of the third degree class. This means that the Stieltjes function associated with the corresponding moment sequence satisfies a cubic equation with polynomial coefficients. We introduce the notion of a primitive triple of a strict third degree form. A simplification criterion of the corresponding cubic algebraic equation is given. Moreover, we show that the class of third degree linear forms is closed under rational spectral transformations. Several consequences of this fact are deduced. In particular, we illustrate with several examples the set of third degree linear forms is stable for the most standard algebraic operations in the linear space of linear forms.

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Acknowledgements

The authors thank the anonymous referees for their corrections and comments which have contributed to improve the presentation of the manuscript. The work of the second author (Francisco Marcellán) has been supported by Agencia Estatal de Investigación (AEI) of Spain, grant PGC2018-096504-B-C31 and FEDER.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, 5019, Monastir, Tunisia

    Imed Ben Salah & Mohamed Khalfallah

  2. Departamento de Matemáticas, Universidad Carlos III de Madrid, 28911, Leganés, Spain

    Francisco Marcellán

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  1. Imed Ben Salah
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  2. Francisco Marcellán
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  3. Mohamed Khalfallah
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Correspondence to Imed Ben Salah.

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Salah, I.B., Marcellán, F. & Khalfallah, M. Stability of Third Degree Linear Functionals and Rational Spectral Transformations. Mediterr. J. Math. 19, 155 (2022). https://doi.org/10.1007/s00009-022-02098-z

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  • DOI: https://doi.org/10.1007/s00009-022-02098-z

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