On functionals which are orthogonally additive modulo \(\mathbb{Z}\). (English) Zbl 0862.39012
The Cauchy difference \(C_f(x,y):=f(x+y)-f(x)-f(y)\) is studied for functions \(f\) mapping a real inner product space \(E\) \((\dim E>1)\) into the reals. The functions satisfying the congruence \(C_f(x,y)\in\mathbb{Z}\) (the set of integers) for all orthogonal \(x,y\in E\) are characterized by means of the square of the norm in \(E\) and the unique real linear functional acting on \(E\).
This is an analogue, obtained both for sets with Baire property and for Christensen measurable sets, of a result by K. Baron and G. L. Forti [Result. Math. 26, No. 3-4, 205-210 (1994; Zbl 0828.39010)]. Following the pattern from the paper just quoted, the author also supplies solutions \(F:E\to \mathbb{C}\) of the conditional Cauchy equation \(F(x+y)=F(x)F(y)\) for all orthogonal \(x,y\in E\).
This is an analogue, obtained both for sets with Baire property and for Christensen measurable sets, of a result by K. Baron and G. L. Forti [Result. Math. 26, No. 3-4, 205-210 (1994; Zbl 0828.39010)]. Following the pattern from the paper just quoted, the author also supplies solutions \(F:E\to \mathbb{C}\) of the conditional Cauchy equation \(F(x+y)=F(x)F(y)\) for all orthogonal \(x,y\in E\).
Reviewer: B.Choczewski (Kraków)
MSC:
| 39B52 | Functional equations for functions with more general domains and/or ranges |
| 39B22 | Functional equations for real functions |
Keywords:
orthogonally additive linear functionals; Cauchy difference; inner product space; congruence; Christensen measurable sets; conditional Cauchy equationCitations:
Zbl 0828.39010References:
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| [8] | M. Kuczma, An introduction to the theory of functional equations and inequalities. [Prace Naukowe Uniwersytetu Slaskiego w Katowicach, Nr. 489]. Panstwowe Wydawnictwo Naukowe – Uniwersytet Slaski, Warszawa – Krakow – Katowice, 1985. · Zbl 0555.39004 |
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