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 | References | Similar Articles | Additional Information

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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Additional Information

Keywords:
Pexider equation,
d’Alembert equation,
convolution type functional equations,
additive map,
exponential map,
quadratic entropy

Article copyright:
© Copyright 1995
American Mathematical Society