On a multiplicative type sum form functional equation and its role in information theory

Prem Nath; Dhiraj Kumar Singh

Applications of Mathematics (2006)

  • Volume: 51, Issue: 5, page 495-516
  • ISSN: 0862-7940

Abstract

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In this paper, we obtain all possible general solutions of the sum form functional equations ∑ i = 1 k ∑ j = 1 ℓ f ( p i q j ) = ∑ i = 1 k g ( p i ) ∑ j = 1 ℓ h ( q j ) and ∑ i = 1 k ∑ j = 1 ℓ F ( p i q j ) = ∑ i = 1 k G ( p i ) + ∑ j = 1 ℓ H ( q j ) + λ ∑ i = 1 k G ( p i ) ∑ j = 1 ℓ H ( q j ) valid for all complete probability distributions ( p 1 , ... , p k ) , ( q 1 , ... , q ℓ ) , k ≥ 3 , ℓ ≥ 3 fixed integers; λ ∈ ℝ , λ ≠ 0 and F , G , H , f , g , h are real valued mappings each having the domain I = [ 0 , 1 ] , the unit closed interval.

How to cite

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Nath, Prem, and Singh, Dhiraj Kumar. "On a multiplicative type sum form functional equation and its role in information theory." Applications of Mathematics 51.5 (2006): 495-516. <http://eudml.org/doc/33264>.

@article{Nath2006,
abstract = {In this paper, we obtain all possible general solutions of the sum form functional equations \[ \begin\{@align\}\{1\}\{-1\}\sum \_\{i=1\}^\{k\}\sum \_\{j=1\}^\{\ell \}f(p\_iq\_j)=&\sum \_\{i=1\}^\{k\}g(p\_i) \sum \_\{j=1\}^\{\ell \}h(q\_j)\\ \text\{and\} \sum \_\{i=1\}^\{k\}\sum \_\{j=1\}^\{\ell \}F(p\_iq\_j)=&\sum \_\{i=1\}^\{k\} G(p\_i)+\sum \_\{j=1\}^\{\ell \}H(q\_j)+ \lambda \sum \_\{i=1\}^\{k\}G(p\_i)\sum \_\{j=1\}^\{\ell \}H(q\_j) \end\{@align\}\] valid for all complete probability distributions $(p_1,\ldots ,p_k)$, $(q_1,\ldots ,q_\ell )$, $k\ge 3$, $\ell \ge 3$ fixed integers; $\lambda \in \mathbb \{R\}$, $\lambda \ne 0$ and $F$, $G$, $H$, $f$, $g$, $h$ are real valued mappings each having the domain $I=[0,1]$, the unit closed interval.},
author = {Nath, Prem, Singh, Dhiraj Kumar},
journal = {Applications of Mathematics},
keywords = {sum form functional equation; additive function; multiplicative function; sum form functional equation; additive function; multiplicative function; real functions; general solutions},
language = {eng},
number = {5},
pages = {495-516},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On a multiplicative type sum form functional equation and its role in information theory},
url = {http://eudml.org/doc/33264},
volume = {51},
year = {2006},
}

TY - JOUR
AU - Nath, Prem
AU - Singh, Dhiraj Kumar
TI - On a multiplicative type sum form functional equation and its role in information theory
JO - Applications of Mathematics
PY - 2006
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 51
IS - 5
SP - 495
EP - 516
AB - In this paper, we obtain all possible general solutions of the sum form functional equations \[ \begin{@align}{1}{-1}\sum _{i=1}^{k}\sum _{j=1}^{\ell }f(p_iq_j)=&\sum _{i=1}^{k}g(p_i) \sum _{j=1}^{\ell }h(q_j)\\ \text{and} \sum _{i=1}^{k}\sum _{j=1}^{\ell }F(p_iq_j)=&\sum _{i=1}^{k} G(p_i)+\sum _{j=1}^{\ell }H(q_j)+ \lambda \sum _{i=1}^{k}G(p_i)\sum _{j=1}^{\ell }H(q_j) \end{@align}\] valid for all complete probability distributions $(p_1,\ldots ,p_k)$, $(q_1,\ldots ,q_\ell )$, $k\ge 3$, $\ell \ge 3$ fixed integers; $\lambda \in \mathbb {R}$, $\lambda \ne 0$ and $F$, $G$, $H$, $f$, $g$, $h$ are real valued mappings each having the domain $I=[0,1]$, the unit closed interval.
LA - eng
KW - sum form functional equation; additive function; multiplicative function; sum form functional equation; additive function; multiplicative function; real functions; general solutions
UR - http://eudml.org/doc/33264
ER -

References

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