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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On a quadratic-trigonometric functional equation and some applications


Authors: J. K. Chung, B. R. Ebanks, C. T. Ng and P. K. Sahoo
Journal: Trans. Amer. Math. Soc. 347 (1995), 1131-1161
MSC: Primary 39B52; Secondary 39B22, 39B32
DOI: https://doi.org/10.1090/S0002-9947-1995-1290715-0
Erratum: Trans. Amer. Math. Soc. 349 (1997), 4691.
MathSciNet review: 1290715
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Abstract: Our main goal is to determine the general solution of the functional equation \[ \begin {array}{*{20}{c}} {{f_1}(xy) + {f_2}(x{y^{ - 1}}) = {f_3}(x) + {f_4}(y) + {f_5}(x){f_6}(y),} \\ {{f_i}(txy) = {f_i}(tyx)\qquad (i = 1,2)} \\ \end {array} \] where ${f_i}$ are complex-valued functions defined on a group. This equation contains, among others, an equation of H. Swiatak whose general solution was not known until now and an equation studied by K.S. Lau in connection with a characterization of Rao’s quadratic entropies. Special cases of this equation also include the Pexider, quadratic, d’Alembert and Wilson equations.


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Keywords: Pexider equation, d’Alembert equation, convolution type functional equations, additive map, exponential map, quadratic entropy
Article copyright: © Copyright 1995 American Mathematical Society