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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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

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
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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