On Popoviciu-Ionescu functional equation

Jose M. Almira

Published: 2016-09-23

Vol. 30 (2016)

On Popoviciu-Ionescu functional equation

Jose M. Almira

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

Download files

PDF

Citation rules

Almira, J. M. (2016). On Popoviciu-Ionescu functional equation. Annales Mathematicae Silesianae, 30, 5–15. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/13953

Similar Articles

Youssef Aissi, Driss Zeglami, Brahim Fadli, Alienation of Drygas' and Cauchy's functional equations , Annales Mathematicae Silesianae: Vol. 35 No. 2 (2021)
Ali Khalouta, Abdelouahab Kadem, Numerical comparison of FNVIM and FNHPM for solving a certain type of nonlinear Caputo time-fractional partial differential equations , Annales Mathematicae Silesianae: Vol. 34 No. 2 (2020)
Wojciech Surówka, A discrete form of Jordan curve theorem , Annales Mathematicae Silesianae: Vol. 7 (1993)
Youssef Aissi, Driss Zeglami, New Pexiderizations of Drygas’ functional equation on abelian semigroups , Annales Mathematicae Silesianae: Vol. 37 No. 2 (2023)
Elhoucien Elqorachi, Ahmed Redouani, Solutions and stability of generalized Kannappan’s and Van Vleck’s functional equations , Annales Mathematicae Silesianae: Vol. 32 (2018)
Zofia Muzyczka, On functional-differential equations with advancing argument , Annales Mathematicae Silesianae: Vol. 2 (1986)
Janusz Brzdęk, On some conditional functional equations of Gołąb-Schinzel type , Annales Mathematicae Silesianae: Vol. 9 (1995)
Harald Fripertinger, Ludwig Reich, Families of commuting formal power series and formal functional equations , Annales Mathematicae Silesianae: Vol. 35 No. 1 (2021)
Youssef Aissi, Driss Zeglami, Mohamed Ayoubi, A variant of d’Alembert’s matrix functional equation , Annales Mathematicae Silesianae: Vol. 35 No. 1 (2021)
Barbara Sobek, Alienation of the Jensen, Cauchy and d’Alembert equations , Annales Mathematicae Silesianae: Vol. 30 (2016)

1 2 3 4 5 6 7 8 9 10 > >>

You may also start an advanced similarity search for this article.

References

1. Aksoy A., Almira J.M., On Montel and Montel–Popoviciu theorems in several variables, Aequationes Math. 89 (2015), 1335–1357.
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. Google Scholar

Vol. 30 (2016)
Published: 2016-09-23

ISSN: 0860-2107

eISSN: 2391-4238

10.1515/amsil

Publisher

University of Silesia Press

Submit