Abstract
Letf, g t ,h t be real or complex-valued functions satisfying the functional equation
for all nonnegativex, y. Supposing thatg 1, ...,g N andh 1, ...,h N are linearly independent on (0, ∞) and [0, ∞) respectively, we prove thatf, g t ,h t (t=1, ...,N) can be extended to the whole real line preserving the functional equation, continuity and measurability.
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References
Aczél, J.,Lectures on functional equations and their applications. Academic Press, New York-London, 1966.
Aczél, J.,Diamonds are not the Cauchy extensionist's best friend. C.R. Math. Rep. Acad. Sci. Canada5 (1983), 259–264.
Aczél, J. andErdös, P.,The non-existence of a Hamel-basis and the general solution of Cauchy's functional equation for nonnegative numbers. Publ. Math. Debrecen12 (1965), 259–263.
Aczél, J. andDaróczy, Z.,On measures of information and their characterizations. Academic Press, New York-San Francisco-London, 1975.
Aczél, J. andChung, J. K.,Integrable solutions of functional equations of a general type. Studia Sci. Math. Hungar.17 (1982–84) 51–67.
Járai, A.,A remark to a paper of J. Aczél and J. K. Chung. To appear.
Daróczy, Z. andLosonczi, L.,Über die Erweiterung der auf einer Punktmenge additiven Funktionen. Publ. Math. Debrecen14 (1967), 239–245.
Kannappan, Pl.,On a generalization of sum form functional equation I. To appear in Publ. Elektrotehn. Fak. Univ. Beograd.
Kannappan, Pl.,On a generalization of sum form functional equation III. Demonstratio Math.13 (1980), 749–754.
Kuczma, M. andZajtz, A,Über die multiplikative Cauchysche Funktionalgleichung fur Matrizen dritter Ordnung. Arch. Math.15 (1964), 136–143.
Losonczi, L.,Functional equations of sum form. InReport of the 20th International Symposium on Functional Equations (Oberwolfach, 1982), Aequationes Math. 26 (1983), 275–276.
Losonczi, L.,Functional equations of sum form. To appear in Publ. Math. Debrecen.
Losonczi, L.,A characterization of entropies of degree α. Metrika28 (1981), 237–244.
Losonczi, L. andMaksa, Gy.,On some functional equations of the information theory. Acta Math. Acad. Sci. Hung.39 (1982), 73–82.
Losonczi, L. andMaksa, Gy.,The general solution of a functional equation of the information theory. Glasnik Mat. Ser. III16 (36) (1981), 261–268.
McKiernan, M. A.,Equations of the form H(x∮y)=Σ i f i (x)g i (x). Aequationes Math.16 (1977), 51–58.
Rimán, J.,On an extension of Pexider's equation. InSymposium en Quasigroupes et Équations Fonctionnelles (Belgrade-Novi Sad, 1974), Zbornik Rad. Mat. Inst. Beograd (N.S.)1 (9) (1976), 65–72.
Székelyhidi, L.,Functional equations on Abelian groups. Acta Math. Acad. Sci. Hung.37 (1981), 235–243.
Székelyhidi, L.,The general representation of an additive function on an open point set (Hungarian). Magyar Tud. Akad. Mat. Fiz. Oszt. Közl.21 (1973), 503–509.
Székelyhidi, L.,An extension theorem for a functional equation. Publ. Math. Debrecen28 (1981), 275–279.
Vincze, E.,Eine allgemeinere Methode in der Theorie der Funktionalgleichungen I, II, III, IV. Publ. Math. Debrecen9 (1962), 149–163,9 (1962), 314–323,10 (1963), 191–202,10 (1963), 283–318.
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Dedicated to Professor János Aczél on his 60th birthday
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Losonczi, L. An extension theorem. Aeq. Math. 28, 293–299 (1985). https://doi.org/10.1007/BF02189422
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DOI: https://doi.org/10.1007/BF02189422