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
Carleson L., “On universal moment problems,” Math. Scand., vol. 9, no. 1, 197–206 (1961).
Carleman T., Les Fonctions Quasi Analytiques, Gauthier-Villars, Paris (1926).
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).
Dzhanashiya G. A., “The Carleman problem in the Jeffrey class of functions,” Dokl. Akad. Nauk SSSR, vol. 145, no. 2, 259–262 (1962).
Bronshtein M. D., “Continuation of functions in nonquasianalytic Carleman classes,” Izv. Vyssh. Uchebn. Zaved. Mat., vol. 12, 10–12 (1986).
Borel E., “Sur quelques points de la theorie des fonctions,” Ann. Ecole Norm. Sup., vol. 3, no. 12, 9–55 (1895).
Björck G., “Linear partial differential operators and generalized distributions,” Ark. Mat., vol. 6, no. 4, 351–407 (1966).
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).
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).
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).
Abanina D. A., “On Borel’s theorem for spaces of ultradifferentiable functions of mean type,” Result. Math., vol. 44, no. 3, 195–213 (2003).
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).
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).
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).
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).
Polyakova D. A., “On solutions of convolution equations in spaces of ultradifferentiable functions,” St. Petersburg Math. J., vol. 26, no. 6, 949–963 (2015).
Boas R. P., Entire Functions, Academic, New York (1954).
Taylor B. A., “On weighted polynomial approximation of entire functions,” Pacific J. Math., vol. 36, no. 2, 523–539 (1971).
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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