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On the Moment Problem in the Spaces of Ultradifferentiable Functions of Mean Type

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Abstract

We consider a version of the classical moment problem in the Beurling and Roumieu spaces of ultradifferentiable functions of mean type on the real axis. We obtain the necessary and sufficient conditions for the weights \( \omega \) and \( \sigma \) under which, for each number sequence in the space generated by \( \sigma \), there is an \( \omega \)-ultradifferentiable function whose derivatives at zero coincide with the elements of the sequence.

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References

  1. Carleson L., “On universal moment problems,” Math. Scand., vol. 9, no. 1, 197–206 (1961).

    Article  MathSciNet  Google Scholar 

  2. Carleman T., Les Fonctions Quasi Analytiques, Gauthier-Villars, Paris (1926).

    MATH  Google Scholar 

  3. Mityagin B. S., “An infinitely differentiable function with the values of its derivatives given at a point,” Dokl. Akad. Nauk SSSR, vol. 138, no. 2, 289–292 (1961).

    MathSciNet  Google Scholar 

  4. Dzhanashiya G. A., “The Carleman problem in the Jeffrey class of functions,” Dokl. Akad. Nauk SSSR, vol. 145, no. 2, 259–262 (1962).

    MathSciNet  Google Scholar 

  5. Bronshtein M. D., “Continuation of functions in nonquasianalytic Carleman classes,” Izv. Vyssh. Uchebn. Zaved. Mat., vol. 12, 10–12 (1986).

    MathSciNet  Google Scholar 

  6. Borel E., “Sur quelques points de la theorie des fonctions,” Ann. Ecole Norm. Sup., vol. 3, no. 12, 9–55 (1895).

    Article  MathSciNet  Google Scholar 

  7. Björck G., “Linear partial differential operators and generalized distributions,” Ark. Mat., vol. 6, no. 4, 351–407 (1966).

    Article  MathSciNet  Google Scholar 

  8. Meise R. and Taylor B. A., “Whitney’s extension theorem for ultradifferentiable functions of Beurling type,” Ark. Mat., vol. 26, no. 2, 265–287 (1988).

    Article  MathSciNet  Google Scholar 

  9. Bonet J., Meise R., and Taylor B. A., “Whitney’s extension theorem for ultradifferentiable functions of Roumieu type,” Proc. Roy. Irish Acad. Sect. A, vol. 89, no. 1, 53–66 (1989).

    MathSciNet  MATH  Google Scholar 

  10. Bonet J., Meise R., and Taylor B. A., “On the range of the Borel map for classes of non-quasianalytic functions,” North-Holland Math. Stud., vol. 170, 97–111 (1992).

    Article  Google Scholar 

  11. Abanina D. A., “On Borel’s theorem for spaces of ultradifferentiable functions of mean type,” Result. Math., vol. 44, no. 3, 195–213 (2003).

    Article  MathSciNet  Google Scholar 

  12. Abanina D. A., “Absolutely representing systems of exponentials with imaginary exponents in ultrajet spaces of normal type and the extension of Whitney functions,” Vladikavkaz. Mat. Zh., vol. 7, no. 1, 3–15 (2005).

    MathSciNet  MATH  Google Scholar 

  13. Abanin A. V. and Pham T. T., “Almost subadditive weight functions form Braun–Meise–Taylor theory of ultradistributions,” J. Math. Anal. Appl., vol. 363, no. 1, 296–301 (2010).

    Article  MathSciNet  Google Scholar 

  14. Zharinov V. V., “Compact families of locally convex topological vector spaces, Fréchet–Schwartz and dual Fréchet–Schwartz spaces,” Russian Math. Surveys, vol. 34, no. 4, 105–143 (1979).

    Article  Google Scholar 

  15. Abanin A. V. and Filipev I. A., “Analytic implementation of the duals of some spaces of infinitely differentiable functions,” Sib. Math. J., vol. 47, no. 3, 397–409 (2006).

    Article  Google Scholar 

  16. Polyakova D. A., “On solutions of convolution equations in spaces of ultradifferentiable functions,” St. Petersburg Math. J., vol. 26, no. 6, 949–963 (2015).

    Article  MathSciNet  Google Scholar 

  17. Boas R. P., Entire Functions, Academic, New York (1954).

    MATH  Google Scholar 

  18. Taylor B. A., “On weighted polynomial approximation of entire functions,” Pacific J. Math., vol. 36, no. 2, 523–539 (1971).

    Article  MathSciNet  Google Scholar 

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Correspondence to D. A. Polyakova.

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Translated from Sibirskii Matematicheskii Zhurnal, 2022, Vol. 63, No. 2, pp. 404–417. https://doi.org/10.33048/smzh.2022.63.211

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Polyakova, D.A. On the Moment Problem in the Spaces of Ultradifferentiable Functions of Mean Type. Sib Math J 63, 336–347 (2022). https://doi.org/10.1134/S0037446622020112

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  • DOI: https://doi.org/10.1134/S0037446622020112

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