Abstract
Our main goal is to introduce some integral-type generalizations of the cosine and sine equations for complex-valued functions defined on a group G that need not be abelian. These equations provide a joint generalization of many trigonometric type functional equations such as d’Alembert’s, Cauchy’s, Gajda’s, Kannappan’s and Van Vleck’s equations. We prove that the continuous solutions for the first type and the central continuous solutions for the second one of these equations can be expressed in terms of characters, additive maps and matrix elements of irreducible, 2-dimensional representations of the group G. So the theory is part of harmonic analysis on groups.
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References
Aczél J.: Lectures on functional equations and their applications. Mathematics in Science and Engineering. Academic Press, New York/London (1966)
D’Alembert, J.: Addition au Mémoire sur la courbe que forme une corde tendue mise en vibration. Hist. Acad. Berlin 1750, 355–360 (1750)
Akkouchi M., Bakali A., Khalil I.: A class of functional equations on a locally compact group. J. Lond. Math. Soc. (2) 57(3), 694–705 (1998)
Badora R.: On a joint generalization of Cauchy’s and d’Alembert’s functional equations. Aequationes Math. 43(1), 72–89 (1992)
Davison T.M.K.: D’Alembert’s functional equation on topological monoids. Publ. Math. Debrecen 75(1–2), 41–66 (2009)
Ebanks B.R., Stetkær H.: On Wilson’s functional equations. Aequationes Math. 89, 339–354 (2015)
Ebanks B.R., Stetkær H.: d’Alembert’s other functional equation. Publ. Math. Debrecen 87(3–4), 319–349 (2015)
Fadli B., Zeglami D., Kabbaj S.: A joint generalization of Van Vleck’s and Kannappan’s equations on groups. Adv. Pure Appl. Math. 6(3), 179–188 (2015)
Gajda Z.: A generalization of d’Alembert’s functional equation. Funkcial. Ekvac. 33(1), 69–77 (1990)
Kannappan Pl.: The functional equation f(xy) + f(xy −1) for groups. Proc. Am. Math. Soc. 19, 69–74 (1968)
Kannappan Pl.: A functional equation for the cosine. Can. Math. Bull. 2, 495–498 (1968)
Kannappan Pl.: Functional equations and inequalities with applications. Springer, New York (2009)
Perkins A.M., Sahoo P.K.: On two functional equations with involution on groups related to sine and cosine functions. Aequationes Math. 89(5), 1251–1263 (2015)
Stetkær H.: Functional equations on abelian groups with involution. Aequationes Math. 54, 144–172 (1997)
Stetkær H.: Functional equations on groups. World Scientific Publishing Company, Singapore (2013)
Stetkær, H.: Van Vleck’s functional equation for the sine. Aequationes Math. (2015a). doi:10.1007/s00010-015-0349-z
Stetkær, H.: Kannappan’s functional equation on semigroups with involution, Semigroup Forum. (2015b). doi:10.1007/s00233-015-9756-7
Székelyhidi L.: Convolution Type Functional Equations on Topological Abelian Groups. World Scientific Publishing Co., Inc, Teaneck (1991)
Van Vleck E.B.: A functional equation for the sine. Ann. Math. Second Ser. 11(4), 161–165 (1910)
Zeglami D., Fadli B., Kabbaj S.: On a variant of μ-Wilson’s functional equation on a locally compact group. Aequationes Math. 89(5), 1265–1280 (2015)
Zeglami, D., Fadli, B., Kabbaj, S.: Harmonic analysis and generalized functional equations for the cosine 7(1), 41–49 (2016)
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Zeglami, D., Fadli, B. Integral functional equations on locally compact groups with involution. Aequat. Math. 90, 967–982 (2016). https://doi.org/10.1007/s00010-016-0417-z
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DOI: https://doi.org/10.1007/s00010-016-0417-z