A Lebesgue Decomposition for Vector Valued Additive Set Functions Defined on a Lattice

1977 ◽  
Vol 29 (2) ◽  
pp. 295-298 ◽  
Author(s):  
Thomas P. Dence
Keyword(s):  
Abelian Group ◽  
Projection Operators ◽  
Additive Functions ◽  
Lebesgue Decomposition ◽  
Set Functions ◽  
Bounded Set ◽  
Vector Valued

Our aim is to establish the Lebesgue decomposition for s-bounded vector valued additive functions defined on lattices of sets in both the finitely and countably additive setting. Strongly bounded (s-bounded) set functions were first studied by Rickart [15], and then by Rao [14], Brooks [1] and Darst [5; 6]. In 1963 Darst [6] established a result giving the decomposition of s-bounded elements in a normed Abelian group with respect to an algebra of projection operators.

scholarly journals A Lebesgue decomposition for elements in a topological group

1980 ◽  
Vol 3 (4) ◽  
pp. 801-808
Author(s):  
Thomas P. Dence
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Projection Operators ◽  
Additive Functions ◽  
Lebesgue Decomposition ◽  
Algebra Of Sets

Our aim is to establish the Lebesgue decomposition for strongly-bounded elements in a topological group. In 1963 Richard Darst established a result giving the Lebesgue decomposition of strongly-bounded elements in a normed Abelian group with respect to an algebra of projection operators. Consequently, one can establish the decomposition of strongly-bounded additive functions defined on an algebra of sets. Analagous results follow for lattices of sets. Generalizing some of the techniques yield decompositions for elements in a topological group.

scholarly journals A Lebesgue decomposition for vector valued additive set functions

1975 ◽  
Vol 57 (1) ◽  
pp. 91-98 ◽  
Author(s):  
Thomas Dence
Keyword(s):  
Lebesgue Decomposition ◽  
Set Functions ◽  
Vector Valued

scholarly journals The lebesgue decomposition for lattices of projection operators

1975 ◽  
Vol 17 (1) ◽  
pp. 30-33
Author(s):  
R.B Darst
Keyword(s):  
Projection Operators ◽  
Lebesgue Decomposition

scholarly journals Optimality of differentiable, vector-valued n-set functions

1990 ◽  
Vol 149 (1) ◽  
pp. 255-270 ◽  
Author(s):  
Lai-Jiu Lin
Keyword(s):  
Set Functions ◽  
Vector Valued ◽  
Differentiable Vector

scholarly journals Duality theorems of vector-valued n-set functions

1991 ◽  
Vol 21 (11-12) ◽  
pp. 165-175 ◽  
Author(s):  
Lai-Jiu Lin
Keyword(s):  
Duality Theorems ◽  
Set Functions ◽  
Vector Valued

Decomposition of vector-valued additive functions

1992 ◽  
Vol 61 (1) ◽  
pp. 1926-1930
Author(s):  
V. N. Sudakov
Keyword(s):  
Additive Functions ◽  
Vector Valued

A special class of functional equations

2018 ◽  
Vol 68 (2) ◽  
pp. 397-404 ◽  
Author(s):  
Ahmed Charifi ◽  
Radosław Łukasik ◽  
Driss Zeglami
Keyword(s):  
Functional Equation ◽  
Finite Group ◽  
Abelian Group ◽  
Special Class ◽  
Functional Equations ◽  
Additive Functions ◽  
Group Of Automorphisms ◽  
Quadratic Equations ◽  
A Finite Group ◽  
Abelian Monoid

Abstract We obtain in terms of additive and multi-additive functions the solutions f, h: S → H of the functional equation $$\begin{array}{} \displaystyle \sum\limits_{\lambda \in \Phi }f(x+\lambda y+a_{\lambda })=Nf(x)+h(y),\quad x,y\in S, \end{array} $$ where (S, +) is an abelian monoid, Φ is a finite group of automorphisms of S, N = | Φ | designates the number of its elements, {aλ, λ ∈ Φ} are arbitrary elements of S and (H, +) is an abelian group. In addition, some applications are given. This equation provides a joint generalization of many functional equations such as Cauchy’s, Jensen’s, Łukasik’s, quadratic or Φ-quadratic equations.

scholarly journals On the Differentiability of Vector Valued Additive Set Functions

2013 ◽  
Vol 03 (08) ◽  
pp. 653-659 ◽  
Author(s):  
Mangatiana A. Robdera ◽  
Dintle Kagiso
Keyword(s):  
Set Functions ◽  
Vector Valued

scholarly journals ON SOME PEXIDER-TYPE FUNCTIONAL EQUATIONS CONNECTED WITH THE ABSOLUTE VALUE OF ADDITIVE FUNCTIONS. PART II

2012 ◽  
Vol 85 (2) ◽  
pp. 202-216 ◽  
Author(s):  
BARBARA PRZEBIERACZ
Keyword(s):  
Functional Equation ◽  
Abelian Group ◽  
Functional Equations ◽  
Additive Functions ◽  
Absolute Value ◽  
Real Functions ◽  
The Absolute ◽  
Image Position

AbstractWe investigate the Pexider-type functional equation where f, g, h are real functions defined on an abelian group G. We solve this equation under the assumptions G=ℝ and f is continuous.

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