On Popoviciu-Ionescu functional equation
Abstract
We study a functional equation first proposed by T. Popoviciu [15] in 1955. It was solved for the easiest case by Ionescu [9] in 1956 and, for the general case, by Ghiorcoiasiu and Roscau [7] and Radó [17] in 1962. Our solution is based on a generalization of Radó’s theorem to distributions in a higher dimensional setting and, as far as we know, is different than existing solutions. Finally, we propose several related open problems.
Keywords
functional equations; exponential polynomials on Abelian groups; Montel type theorem
References
2. Almira J.M., Montel’s theorem and subspaces of distributions which are Δ^m-invariant, Numer. Funct. Anal. Optim. 35 (2014), no. 4, 389–403.
3. Almira J.M., Abu-Helaiel K.F., On Montel’s theorem in several variables, Carpathian J. Math. 31 (2015), 1–10.
4. Almira J.M., Székelyhidi L., Montel-type theorems for exponential polynomials, Demonstratio Math. 49 (2016), no. 2, 197–212.
5. Anselone P.M., Korevaar J., Translation invariant subspaces of finite dimension, Proc. Amer. Math. Soc. 15 (1964), 747–752.
6. Constantinescu F., La solution d’une équation fonctionnelle à l’aide de la théorie des distributions, Acta Math. Acad. Sci. Hungar. 16 (1965), 211–212.
7. Ghiorcoiasiu N., Roscau H., L’integration d’une équation fonctionnelle, Mathematica (Cluj) 4 (27) (1962), 21–32.
8. Hardy G.H., Wright E.M., An Introduction to the Theory of Numbers, Fifth edition, The Clarendon Press, Oxford University Press, New York, 1979.
9. Ionescu D.V., Sur une équation fonctionnelle, Studii si cercet. de mat. Cluj 8 (1956), 274–288.
10. Kuczma M., A survey of the theory of functional equations, Univ. Beograd. Publ. Elektrotchn. Fak. Ser. Mat. Fiz. 130 (1964), 1–64.
11. Levi-Civita T., Sulle funzioni che ammetono una formula d’addizione del tipo $f(x + y) = \sum_{i = 1}^n {X_i (x)Y_i (y)} $, R. C. Accad. Lincei 22 (1913), 181–183.
12. Montel P., Sur quelques extensions d’un théorème de Jacobi, Prace Matematyczno-Fizyczne 44 (1937), no. 1, 315–329.
13. Montel P., Sur quelques équations aux differences mêlées, Ann. Sci. École Norm. Sup. 65 (1948), no. 3, 337–353.
14. Montel P., Sur un système d’équations fonctionnelles, Ann. Soc. Polon. Math. 21 (1948), 99–106.
15. Popoviciu T., Sur quelques équations fonctionnelles, (Romanian) Acad. R. P. Romine. Fil. Cluj. Stud. Cerc. Sti. Ser. I 6 (1955), no. 3–4, 37–49.
16. Prager W., Schwaiger J., Generalized polynomials in one and several variables, Math. Pannon. 20 (2009), no. 2, 189–208.
17. Radó F., Caractérisation de l’ensemble des intégrales des équations différentielles linéaires homogènes à coefficients constants d’ordre donné, Mathematica (Cluj) 4 (27) (1962), 131–143.
18. Shulman E.V., Addition theorems and representations of topological semigroups, J. Math. Anal. Appl. 316 (2006), 9–15.
19. Shulman E.V., Decomposable functions and representations of topological semigroups, Aequationes Math. 79 (2010), no. 1–2, 13–21.
20. Shulman E.V., Some extensions of the Levi-Civita functional equation and richly periodic spaces of functions, Aequationes Math. 81 (2011), no. 1–2, 109–120.
21. Shulman E.V., Subadditive set-functions on semigroups, applications to group representations and functional equations, J. Funct. Anal. 263 (2012), no. 5, 1468–1484.
22. Shulman E.V., Subadditive maps and functional equations, Funct. Anal. Appl. 47 (2013), no. 4, 323–326.
23. Shulman E.V., Addition theorems and related geometric problems of the group representation theory, in: Recent Developments in Functional Equations and Inequalities, Banach Center Publ., Polish Acad. Sci. Inst. Math., Warsaw, 2013, pp. 155–172.
24. Stamate I., Contributii la integrarea unei ecuatii functionale, Inst. Politehn. Cluj, Lucrariti (1960), 47–51.
25. Stetkaer H., Functional equations on groups, World Scientific Publishing Co., Hackensack, 2013.
26. Székelyhidi L., Convolution type functional equations on topological abelian groups, World Scientific Publishing Co., Hackensack, 1991.
Departamento de Matemáticas, Universidad de Jaén, Spain Spain
This work is licensed under a Creative Commons Attribution 4.0 International License.
The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.
- License
This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license. - Author’s Warranties
The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s. - User Rights
Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor. - Co-Authorship
If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.
