Abstract
In this paper, we investigate the solutions of Fréchet’s functional equation in the context of Lie groups. In particular, we give the explicit right-abelian solutions of this equation for connected Lie groups. We also extend this result to homogeneous spaces and deal with some classical examples.
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References
Almira, J.M.: Characterization of polynomials as solutions of certain functional equations. J. Math. Anal. Appl. 459, 1016–1028 (2018)
Almira, J.M., López-Moreno, A.J.: On solutions of the Fréchet functional equation. J. Math. Anal. Appl. 332, 1119–1133 (2007)
Almira, J.M., Shulman, E.V.: On polynomial functions on non-commutative groups. J. Math. Anal. Appl. 458, 875–888 (2018)
Almira, J.M., Székelyhidi, L.: Characterization of classes of polynomial functions. Mediterr. J. Math. 13, 301–307 (2016)
Boole, G., Moulton, J.F.: A Treatise on the Calculus of Finite Differences, 2nd edn. Dover (1960)
Djoković, D.Ž.: A representation theorem for \((x_1-1)(x_2-1)...(x_n-1)\) and its applications. Ann. Polon. Math. 22, 189–198 (1969)
Fréchet, M.: Une définition fonctionnelle des polynômes. Nouv. Ann. 9, 145–162 (1909)
Fréchet, M.: Les polynômes abstraits. J. Math. Pures Appl. 9(8), 71–92 (1929)
Hamel, G.: Einer Basis aller Zahlen und die unstetigen Lösungen der Funktionalgleichung \(f(x+y)= f(x) +f(y)\). Math. Ann. 60, 459–472 (1905)
Jordan, C.: Calculus of Finite Differences, 3rd edn. AMS Chelsea Publishing (1965)
Koh, E.L.: The Cauchy functional equation in distribution. Proc. Am. Math. Soc. 106, 641–646 (1989)
Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities, 2nd edn. Birkhäuser (2009)
Leibman, A.: Polynomial mappings of groups. Israel J. Math. 129, 29–60 (2002)
Millsaps, K.: Abstract polynomials in non-abelian groups. Bull. Am. Math. Soc. 49, 253–257 (1943)
Molla, A., Nicolay, S., Schneiders, J.-P.: On some generalisations of the Fréchet functional equations. J. Math. Anal. Appl. 466, 1400–1409 (2018)
Popa, D., Raşa, I.: The Fréchet functional equation with application to the stability of certain operators. J. Approx. Theory 164, 138–144 (2012)
Prager, W., Schwaiger, J.: Generalized polynomials in one and in several variables. Math. Pannon. 20, 189–208 (2009)
Shulman, E.V.: Each semipolynomial on a group is a polynomial. J. Math. Anal. Appl. 479, 765–772 (2019)
Stetkær, H.: Functional Equations on Groups. World Scientific, Singapore (2013)
Stetkær, H.: Kannappan’s functional equation on semigroups with involution. Semigroup Forum 94, 17–30 (2017)
Székelyhidi, L.: Fréchet’s equation and Hyers theorem on noncommutative semigroups. Ann. Polon. Math. 48, 183–189 (1988)
Van der Lijn, G.: La définition fonctionnelle des polynômes dans les groupes abéliens. Fund. Math. 33, 42–50 (1939)
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Funding was provided by Fonds pour la Formation à la Recherche dans l’Industrie et dans l’Agriculture (Grant no. 35484559)
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Molla, A., Nicolay, S. The Fréchet Functional Equation for Lie Groups. Mediterr. J. Math. 18, 68 (2021). https://doi.org/10.1007/s00009-021-01701-z
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DOI: https://doi.org/10.1007/s00009-021-01701-z