On a functional equation connected with Rao’s quadratic entropy
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- by J. K. Chung, B. R. Ebanks, C. T. Ng and P. K. Sahoo
- Proc. Amer. Math. Soc. 120 (1994), 843-848
- DOI: https://doi.org/10.1090/S0002-9939-1994-1180464-6
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Abstract:
We determine the general solution of the functional equation fxy, \[ f\left ( {\frac {{x + y}} {2}} \right ) + f\left ( {\frac {{x - y}} {2}} \right ) = 2f\left ( {\frac {x} {2}} \right ) + 2f\left ( {\frac {y} {2}} \right ) + \lambda f(x)f(y),\] /: [-$f:[ - 1,1] \to {\mathbf {R}}$. This equation was used by Lau in order to characterize Rao’s quadratic entropies. The general solution is obtained here as a special case of a more general result for $f$ mapping a neighborhood of $0$ in linear topological space into a field.References
- János Aczél, The general solution of two functional equations by reduction to functions additive in two variables and with the aid of Hamel bases, Glasnik Mat.-Fiz. Astronom. Društvo Mat. Fiz. Hrvatske Ser. II 20 (1965), 65–73 (English, with Serbo-Croatian summary). MR 198023 J. K. Chung, B. R. Ebanks, C. T. Ng, and P. K. Sahoo, On a quadratic-trigonometric functional equation and some applications, submitted.
- Ka-Sing Lau, Characterization of Rao’s quadratic entropies, Sankhyā Ser. A 47 (1985), no. 3, 295–309. MR 863724
- László Székelyhidi, Convolution type functional equations on topological abelian groups, World Scientific Publishing Co., Inc., Teaneck, NJ, 1991. MR 1113488, DOI 10.1142/1406
Bibliographic Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 120 (1994), 843-848
- MSC: Primary 39B22
- DOI: https://doi.org/10.1090/S0002-9939-1994-1180464-6
- MathSciNet review: 1180464